Happy new year Emojis & Symbols

⠀⠀⠀⣶⠖⠀⠀⠀⠀⠀⠀⠀⠲⣶⣶⣶⣶⠖⠀⠀⠀⠀⠀⠀⠀⠰⣶⣶⣶⣶⡖⠀⠀⠀⠀⠀⠀⠀⠲⣶⣶⣶⣶⣶
⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡏⠀⠀⠀⠀⣀⡀⠀⠀⠀⠘⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠆⠀⠀⠀⢀⣿⣿⠃⠀⠀⠀⢸⣿⣷⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⠂⠀⠀⠀⢰⣿
⠀⠀⠀⣧⠀⠀⢀⣾⠏⠀⠀⠀⠀⣼⣿⣿⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⣿⣿⣧⡀⠀⢀⣼⠟⠀⠀⠀⠀⣸⣿⣿⣿⣿⣧⣄⡀⠀⠀⠀⠙
⠀⠀⠀⣿⣷⣤⣾⠏⠀⠀⠀⠀⣼⣿⣿⣿⡀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⣿⣿⣿⣷⣤⣾⠏⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⣿⣿⣿⠃⠀⠀⠀⠀⠈⠙⠙⣿⣿⡇⠀⠀⠀⠘⠿⠏⠀⠀⠀⢀⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠋⠉⣿⣿⣿⣿⠛⠿⠋
⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠒⠀⠀⠀⠀⠀⠐⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠀⠀⠀⠀⠀⠒⠛
⠀⠀⠀⢸⣇⣿⡇⢰⡿⣿⡆⣴⡿⣿⡆⣾⣿⣷⠹⣷⣾⠃⠀⢠⣾⢿⣦⢰⣿⠿⢰⣿⢸⡇⣿⡇⠀⠘⣷⣾⠏⣾⡿⠇⣰⡿⣿⡆⣾⡿⣿⡆
⠀⠀⠀⣸⡟⣿⡇⢸⡷⣿⡇⣿⡟⠛⠁⣿⡟⠋⠀⢸⡇⠀⠀⢸⣿⢸⣿⢸⣿⣿⠸⣿⣼⣧⣿⡇⠀⠀⢸⡇⠀⣿⣿⡇⣿⡷⣿⡇⣿⡿⣿⡅
⠀⠀⠀⠈⠁⠉⠁⠈⠁⠉⠁⠉⠁⠀⠀⠉⠁⠀⠀⠈⠁⠀⠀⠈⠉⠈⠉⠀⠉⠉⠀⠉⠉⠉⠉⠀⠀⠀⠈⠁⠀⠈⠉⠁⠉⠁⠉⠁⠉⠁⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣾⣿⣧
⠀⠀⠀⠀⠀⠀⣀⣀⠭⠟⢁⣠⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠵⠶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣭⣭⣅⣀⣀⡀
⠀⠀⠀⠀⠀⢤⣄⣉⡩⠭⠛⠁⣀⡄⠀⠀⠀⠀⠀⠐⣿⡤⠶⠶⠿⠛⠃⠀⠀⠀⠀⠀⠀⣤⣴⣶⣶⣒⣛⣛⣋⣁⣀
⠀⠀⠀⢠⣀⡀⠉⠙⠓⠚⣋⡭⠋⠀⠀⠀⠀⠀⠀⣾⣁⣤⣤⣴⣶⣶⣶⠄⠀⠀⠀⠀⢀⣤⣭⣿⣿⣿⣿⣿⡿⠛⠛⠃
⠀⠀⠀⠀⠑⠬⣉⡘⠉⠉⣀⡠⠄⠀⠀⠀⠀⠀⣴⠋⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣨⣭⣭⣽⣿⣿⣿⣿⠿⠿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠈⢩⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠶⠶⢶⠶⠶⠶⠖⠂⠀⠀⠀⠀⠉⠙⠛⠛⠛⣛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣶⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠿⠿⠆
⠀⠀⠀⠀⠀⣠⡴⠾⠲⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⢷⣶⡶⠶⠋⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣉⣉⣭⠵⠶⠆
⠀⠀⠀⠀⢀⣹⠶⣶⣶⣶⣶⣶⡄⠀⠀⠀⠀⠀⠘⣷⣦⣀⡀⠉⠛⠛⢛⣿⠏⠀⠀⢀⣄⠀⠀⠀⠉⠙⢛⣛⣒⣒⣲⠶⠄
⠀⠀⠀⡞⠉⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢯⣝⣛⠛⠛⠋⢁⣀⣤⠶⣻⠇⠀⠀⠀⠚⠛⠯⠥⠤⣤⣤⣄⡀
⠀⠀⠀⣹⡶⣶⣛⣛⣛⠛⠛⠛⠲⣦⡀⠀⠀⠘⢷⣄⠀⠀⠈⠉⠛⠛⠛⠋⠉⢀⡴⠋⠀⠀⠐⠚⠛⠛⠛⠛⠛⠛⠒⢲⣤
⠀⠀⣸⠁⠀⠀⠈⠉⠉⠉⠛⠛⠋⠉⠀⠀⠀⠀⠀⠙⢿⣶⣤⣤⣤⣤⣤⠴⠞⠋⠀⠀⠀⣀⠀⠀⣀⣠⣤⡶⠒⠚⠉⠉
⠀⠀⢹⣶⣶⣛⣋⣉⣉⣉⣉⣉⣛⣛⣶⠄⠀⢀⡀⠀⠀⠙⠳⢶⣤⣤⣀⣠⣤⣤⣴⣶⠟⠁⠀⠀⠉⠉⠉⠉⠉⣉⣉⣹⠷⠆
⠀⠀⠻⣄⡀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠈⠛⠷⢶⣤⣤⣤⣤⣤⣤⠶⠾⠛⠉⠀⢀⣀⣤⡤⠶⠒⠚⠋⠉⠉
⠀⠀⠀⠈⠙⠛⠛⠛⣿⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⣉⣛⣻⣿⠷⠶⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⡀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⢀⣠⣤⣴⣶⣒⣫⣭⣭⣤⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡌⡜⣰⣵⡗⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⢀⠀⡴⢠⢆⡞⣼⣼⠿⠏⠐⠂⠀⠀⠀⠀⢀⣠⣀⠀⠀⡀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⢀⡞⣼⢡⣯⣾⣿⣯⡭⠭⠅⠁⠀⠀⠀⠀⢀⣸⣤⠿⠀⣼⠑⢛⡷⠀⠀⣠⡾⠀⠀⡴⠱⢛⣿
⠀⠀⠀⠺⢞⣋⣛⣓⡒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠐⠛⠻⠶⠛⠀⠙⠛⠋⠁⠐⠛⠿⠶⠂⠀⠛⠚⠛⠁

⣿⣿⣿⣻⣛⣛⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣛⣻⣟⣿⣟⣻⣿⣛⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⠀⠀⠠⠀⠀⠄⠀⠄⠀⢀⠀⠀⠀⠂⢀⠀⠀⠀⠀⠀⠠⠀⠂⡀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠁⠀⠀⠀⠀⠄⠀⠀⠐⠀⠀⠀⠀⠀⠁⠀⠀⢀⠀⠀⠂⠰⠀⠀⠀⠀⠀⠂⠀⠂⠀⠀⠐⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠒⢤⡀⠀⠈⠀⠀⠀⠀⠀⠒⠂⡀⠀⠀⠀⠉⠀⠉⠒⠠⠈⠀⠠⡂⠀⠀⠀⠈⠐⠠⡀
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⢳⡈⠀⡀⠀⢀⡤⣄⠀⠀⣧⠀⡀⠀⣠⠤⡄⠀⠈⣇⠀⢠⠀⢀⡤⠤⣄⠀⠀⣿⠐⠈⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⡏⠙⠃⠀⠈⡇⠀⠃⠀⣾⠠⠘⠀⠀⢿⠀⣇⣀⡏⠀⡇⠀⠀⣿⠀⠀⠀⢸⡇⠀⠈⠀⠀⢸⠀⠀⠃⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠒⠚⠂⠀⠀⠀⠀⡇⣰⠀⠀⣿⠈⡀⠀⠀⢸⠀⠍⠉⠁⡀⠁⠀⠀⡿⢀⠀⠀⢸⡃⠀⠀⠀⠀⢸⠀⠂⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⢸⠇⢸⠀⠀⣿⠀⠀⠀⡀⢸⠀⠂⠅⠘⠁⠀⢀⡼⠃⠘⠀⠀⢸⡇⠀⡐⠀⠀⢸⠐⠀⠂⠀⠁⠀⠀⠄
⠀⠀⠀⠀⠀⠁⡀⠐⠀⠀⠀⢀⠀⠔⠀⠀⣠⠟⢠⠘⠄⠄⣿⠀⠀⠀⡀⢸⠀⠂⢠⠂⣤⠞⠉⠠⠈⠀⠀⠁⢸⡇⠀⢰⠈⠀⣸⠀⠀⠀⠀⡈⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡎⣵⠋⠀⠀⠁⠠⠀⠀⢸⠀⠀⠀⠄⡟⠀⠀⠀⣾⠁⠀⠀⠀⠈⠀⠀⠀⠸⡇⠁⠀⠀⡀⣸⠀⠀⠀⠀⢀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠛⠒⠒⠢⡆⠈⣃⠠⠈⢓⡂⠚⢠⠇⠸⡄⠀⠙⠉⠁⠀⠉⣮⠰⠆⡀⠀⠛⠒⢖⡪⢁⡟⠀⠀⠀⠠⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠠⠀⠠⠀⠀⠀⣤⣥⣬⣬⣥⣤⡇⠐⠈⠳⠦⣤⡤⠴⠏⠈⠀⣧⣤⣤⣤⣤⣤⣤⣿⢀⠈⠻⠦⣤⣤⡤⡴⠞⠁
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠈⠀⠀⠠⠀⠐⠀⠀⢀⣀⠀⣰⡀⢀⠠⢐⠀⠀⡠⠀⠀⠀⠀⠀⠠⠀⠀⡀⠀⣠⠀⠀⠀⠀⠠⠄⡆⠀⠰⠄⠥⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠆
⠀⠀⠀⠀⢀⠁⠀⠠⠀⢀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠠
⠀⠀⠀⠐⢀⠀⠀⠀⠠⠀⠀⠀⠀⠄⢀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠐⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⢀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⡟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
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⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⣾⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡋⠛
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠋
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠻⣟⣿⣿⣿⣿⣿⠃⠀⠀⠀⢀⣠⣤⠀⠀⢀⣠⠾⠋
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣦⣼⡿⠀⢈⣿⣿⣿⣿⣶⣾⠟⠋⠉⢛⣷⣟⠛⣠⡐
⡇⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣁⢙⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣯⣀⣠⣴⣿⣿⣿⡟⠃
⡇⠀⠀⠀⠀⠀⢰⣿⡉⠰⢶⣤⣀⣀⡀⠀⠀⠀⠀⣠⣤⣴⣆⡙⠿⣿⣿⣿⣷⡄⠲⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡇⠀⠀⠀⠀⠀⣿⡏⠀⠀⠈⠉⠉⠻⢿⣶⣶⣄⣀⣠⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⡇⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⡿⠋⠙⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⢹⣿⣿⣿⣿⣿⣿⡇
⡇⠀⠀⠀⠀⣼⣿⡇⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣼⣆⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣷
⡇⠀⠀⠀⠀⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⣿⣿⣿⣿⣿⣿⣿⡇⢈⣿⣿⣿⣷
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠦
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⠟
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣤⣄⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⠟⢿⣿⣿⣿⣿⡿⢿⣿⣿⣿⡿⠛⠉
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣤⣼⣿⣿⣿⣧⣀⡀⢀⣠⣤⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⣽⠇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⣿⣿⡿⠟⢁⠀⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠉⠀⠀⠀⢸⣿⣿⣿⣟⠉⠉⠙⢿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣯⣿⣡⡆⢸⡏⡿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⠀⠀⢠⣾⣿⣿⠿⢿⢸⣿⢻⣿⣿⣿⡿⣿⠇⣾⢹⠇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⢦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⢠⣴⢿⡿⢿⣯⣄⠄⣸⡇⢸⣿⢸⡟⢠⣿⣸⣧⠚
⡇⠀⠀⠀⠀⠀⠀⠀⢠⠂⠉⠉⢀⠌⢀⣼⠟⣠⠃⠀⠀⡀⠀⣠⢤⡀⣴⠀⣿⣿⣿⣿⣿⠀⠈⠻⣦⡼⢇⣼⠏⠀⠘⠋⠈⠓⠁⠈⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⠤⡾⠀⣻⡷⢚⢥⡄⢰⡞⢹⢠⡏⢸⣧⡟⢸⠃⢽⣿⣿⡈⠓⠶⠶⠶⠞⠛⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⢉⣿⢠⡏⢰⠁⣾⠁⡼⣼⠁⡞⢹⣡⡿⠀⢸⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀
⡇⠀⠀⠀⠀⠤⢤⢴⡀⠀⠀⣿⠀⢸⡇⢸⠁⣜⣰⣯⠸⢁⠟⠚⠀⠀⢡⠃⠀⣿⣿⣿⠋⠠⢀⣠⡤⢄⠀⣠⠞⠉⣱
⡇⠀⠀⡎⠀⠀⠘⠢⠟⢀⡼⠃⠀⠿⠃⠘⠋⠁⠘⠀⠀⡸⠀⠀⢀⡠⠊⠀⢰⣿⣿⣏⡴⠛⢻⣟⡆⠀⢰⣏⣠⠞⠁⠀⢀⠀⣤⡖
⡇⠀⠀⠙⠒⠤⠄⠠⠒⠉⠀⠀⠀⠀⠀⠀⢀⠔⠁⠠⠊⠀⠀⠈⠁⠀⠀⠀⠸⣿⠋⣯⡀⠀⣴⠟⠁⠀⣿⠁⡴⢦⣰⠉⡟⢀⡿⠉
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⡀⠉⠰⡿⠀⠀⣴⣿⢰⡷⢸⡏⢠⡅⣼⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣾⣷⠞⠶⢿⠂⠉⢸⡏⢹⠃⠸⢧⠜⠗⠛
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣿⣁⠈⠷⠒⠊⣁⡴⠋⠀⠈⠉
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡗⠛⠛⠉
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠉⠉⣿⡄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⣏⠀⣿⢈⢳⡄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⣿⡅⠀⡿⣧⠙⠂⣿⠁⢸⡇⣸⠀⢹⡆⢻⣆⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢻⡇⢠⣇⣻⡆⠀⣿⠀⡼⠁⣿⣀⡼⠃⠀⢿⣤⠏⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⡠⠒⠒⢤⠀⠀⠀⣿⡷⠚⢻⡇⢸⠉⠉⣷⠀⣿⠊⠀⠀⣿⡅⠀⠀⠀⢘⣿⠀⠀⠀⠀⠐⠁⣀⠉⢶⡀
⠀⠀⠀⠀⢸⠁⢰⣆⣠⠇⠀⠀⣿⡇⠀⠜⠃⠉⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠠⣄⡉⠙⠳⢦⣀⠀⠐⠀⠛⠀⣼⠃
⠀⠀⠀⠀⠘⢧⡀⠈⠁⣠⡶⠛⠁⠀⢀⣠⣤⣤⣤⣤⣀⣀⠉⢠⡴⠛⠉⠑⢀⡄⠀⠙⣧⠀⠀⠙⠳⠤⢀⠠⠔⠉⣠
⠀⢠⡶⠋⠁⣦⠈⠉⠉⠁⠀⠀⢠⡾⠛⠉⠁⡀⠀⠀⠀⠀⣸⡿⠛⠛⠒⠛⠋⠀⠀⠀⣼⠀⢀⣶⠂⡄⠀⠀⠀⠀⣿
⢰⡿⠁⠀⢰⡿⠀⢠⠒⢺⡆⠀⢾⠀⠀⠀⣰⠇⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠐⠚⠁⠀⠈⠻⣄⣐⢀⣀⠀⢸⡿
⣿⡇⠀⠀⣿⡇⡴⠁⠀⣿⠇⠀⠀⠑⠒⠋⠁⠀⠀⣄⠀⢸⡟⠀⠀⣰⡟⠀⣀⠀⠀⢀⡀⠀⣠⡀⠀⡈⠀⣿⡇⢼⠃
⣿⡇⠀⢸⣿⣸⠁⠀⢰⣿⠀⣠⠤⠀⣠⡆⠀⡀⠀⢸⠀⢸⣇⢀⢜⣿⠇⣾⣁⠈⢠⡟⠀⣴⣿⠀⢠⠃⣼⡿⣴⢢⡀
⠈⠛⠀⢸⣿⠇⠀⠀⣾⡏⣼⡏⠔⣠⣿⠀⣼⡇⠀⡘⠀⠀⠙⠉⣸⡿⠘⣿⢀⠔⠙⠧⠎⠸⡇⡠⠃⠀⢿⡇⢈⣾⠃
⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠻⠧⠞⠁⠻⠔⠈⠓⠊⠀⠠⠦⠤⣤⣿⠁⠀⠈⠀⠀⣤⠤⠀⢄⠀⠀⠠⣄⠀⠉⠉⠁
⠀⠀⠀⠀⠁⠀⠀⠀⠹⣇⠀⠀⢀⡦⠀⢠⠴⠒⠛⠒⠲⠶⣤⣿⡉⠙⠲⠦⠄⢸⡇⠀⡷⢨⠆⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠳⣄⡁⠀⠀⠀⢀⣠⠖⠋⠉⠙⠻⢶⣦⣤⣀⣉⣩⣤⠞⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠆⠀⠀⠈⠀⠀⠀⠈⠀⠀⠁⠀⡅⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠒⡀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠂⠀⠦⡓⢎⡀
⠀⠀⠂⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠆⠀⠀⠀⢁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠘⠈⠂
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠄⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡠⠂⠀⠀⠀⠠⠈⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡀
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠀⠀⠡⠀⢀⠄⢐⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡀⠄
⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠠⠬⠀⡈⡈⠈⡀
⠀⢀⠊⡀⢄⠄⢠⠁⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠀⠀⠀⠤⣈⡠⠀⠀⠀⠠⡈⡠⠀⠀⠀⠀⠀⠀⠀⠁⠠⠐⢈⠀⢑
⠀⡒⠠⠱⠈⡅⠃⢁⠀⠅⠀⠀⡀⠀⠀⠈⠁⠀⠀⠀⠀⠩⠊⠐⠂⠀⠀⠀⠉⠁⠀⠀⠀⠄⠀⠀⠈⠀⢄⠀⠀⠀⠂
⠀⠀⠄⢂⢁⣂⡈⠂⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠈⠀⠌⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠄⠀⠀⠀⠠⠄⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠄⠀⠀⡈⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘
⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⠂⠀⠀⠈⠀⠀⠆⠀⠃⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⢠⡄⠀⠀⠁⠀⠀⠀⠀⠈⠀⠈⠈⠁⠀⠀⠁⠀⠀⠀⠀⡀⠉⠄⠠⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⢾⢗⡁⠀⠀⠀⡄⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⡀⠀⠀⢀⢀⠈⠄⣁⠐⠐⡀⠡
⠀⢀⠀⠁⠀⠄⠀⠀⡀⠐⠀⠀⠀⠘⠀⠀⠀⠀⠥⠬⠀⠀⠀⠀⠀⠥⠌⠀⠀⠀⠀⠀⠀⠀⠀⠂⡂⠠⣈⠱⠠
⠀⠀⢀⠀⠀⠄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠣⡀⠁⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠑⠀⠀⠀⠀⢈⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠠⠈⠂⠀⢀⢀⠀⠀⠀⠀⠀⠂⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⢜⠀⠔⢈⢂⠀⠣⠀⠀⠀⢀⠀⠀⠀⠀⠃⠀⠀⠀⢀⣀⡀⠀⠐⠀⠂⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄
⠀⠀⡀⡈⠐⠊⣠⡔⠀⠑⠀⠈⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢈⠃⠀⠀⠐⠀⠀⠀⢀⠈⠀⡉
⠀⠀⠀⠉⠐⠀⠀⠀⠀⠀⠀⠀⠁⠉⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠈⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠⠐⠈⠐⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⠅⠀⠰⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⣉⠸⡆⠀⠀⠩⠀⠀⠀⠀⠀⠀⠄⠠⠅⠀⠀⠀⠡⢀⢀⡀⠜⠀⠀⠀⡀⠠⠀⠤⠠⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⣉⣤⠍⠀⠀⠀⠡⢀⠀⠂⠀⠀⠠⠈⠀⠀⠀⠀⢀⣀⢀⣴⠀⠀⠀⠄⢩⠐⢂⠉⣐⠃⠤⠒⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣠⡀⠀⠀⠀⠀⠈⠈⠈⠀⣤⡄⢰⣆⡞⣿⢸⡏⡜⣿⡠⢀⠃⠐⠀⠒⡀⠤⢀⡘⢀⢒⡈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⢻⣧⠀⠀⢀⡀⠀⠀⣾⣼⢡⡇⣸⣿⢀⡿⠘⠿⠀⣿⠃⢨⠈⡁⢈⠀⣁⠀⠀⡒⢠⠀⠄⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⣧⡀⠈⣿⡄⠀⣾⢁⣿⢀⣿⠇⠾⠟⢹⣧⠀⠀⠀⣠⠞⣿⠀⠀⠆⠡⠄⠰⢀⡘⢀⠀⢆⠐⣀⠃
⠀⠀⠀⠐⠀⠀⠀⡀⠀⠳⠄⢘⣷⡖⠹⣷⠀⣿⡎⠘⠋⣿⠀⠀⠀⠈⠛⠀⠀⢰⠃⢰⡟⠀⠀⠐⠃⡜⠃⠤⠠⠄⠖⡈⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠹⣷⠀⠘⢧⡀⠀⠀⠀⠻⠀⠀⢠⠖⢻⣧⠀⣼⡟⠋⠀⣀⡀⠀⠀⠀⠈⠀⠂⠘⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⣧⢹⡇⠀⠀⠉⠋⠀⠀⠀⠀⠀⠸⠄⣸⡿⠀⣿⠀⣰⡿⣫⠟⢀⣴⠟⣀⠀⢀⢿⡀⠀⠀⢀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣿⡼⠁⠀⠀⠀⠀⣀⠀⠀⣴⡄⡀⠀⢻⣃⠼⣿⠀⣿⠛⣉⠔⣿⡧⢺⣟⡤⠋⣾⢃⡔⢠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⢹⡄⠀⣰⡿⡆⠀⣿⢸⡇⣸⠖⠁⢀⡄⠀⢠⡿⠀⠈⠉⠁⠀⠀⠀⠀⠁⠀⠸⠷⢋⣀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣷⣄⠈⢧⠀⣿⣇⠇⡸⢿⠆⠑⠋⠀⠀⠸⣄⣠⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡗⠺⣆⠘⢿⡤⠃⠀⠀⠀⢠⢠⢀⠀⠀⠀⠀⠀⢠⠊⣋⢣⡄⠀⠐⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠸⣷⠀⠹⠗⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⠐⡀⠀⠀⠘⣤⣉⣤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠞⠀⡿⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⢐⡀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠛⠁⡠⠀⠄⠄⠀⠀⢐⡀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⢠⠀⠈⡂⠀⠀⠐⠠⢀⠀⢈⠠⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⡀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⣀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡤⠄⠀⢰⣏⣀⡀⢠⡄⠀⢅⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠁⢀⡤⡄⢑⢒⡆⠀⠌⣸⠃⣾⠀⠀⢀⠖⠀⡿⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢋⢠⢱⠀⢠⡇⣿⠀⣾⠐⠁⠂⠸⠅⠊⠀⠈⠈⢤⠀⠀⠈⠑⣪⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠻⣞⡉⣀⠆⠘⠆⠙⠂⠈⠉⣀⠀⣼⠆⠀⠐⠒⠒⠉⢀⠤⠀⠀⡷⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡀⠀⣠⢛⠀⠀⡀⠀⠀⠰⠆⠰⡇⠀⡿⠀⣦⢊⡆⣸⢠⡟⢰⡇⠔⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡼⢣⣄⠁⣿⡈⢀⠘⢤⢴⠇⣾⠀⣀⣿⢨⡇⢸⠇⠸⠀⠻⠀⠉⠈⠀⠎⢹
⠀⠀⠀⠀⠀⢀⠀⢸⡇⠘⠁⢀⡿⠀⡞⡠⠃⢿⠀⠻⠘⡁⣸⠀⠁⠀⣰⡣⠀⠀⠀⠀⢀⡤⠀⢁
⠀⠀⠀⠀⠀⢁⠄⠈⠳⠤⠔⠉⠃⠀⣉⡠⠀⠀⠀⠀⠀⠈⠁⠀⢀⢀⡏⠀⠀⠀⠀⠀⠻⠰⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠠⡀⠀⡴⢂⠀⡶⢠⢃⡏⢸⡇⠀⠀⠀⠀⡴⠀⠀⢄⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡀⠀⠀⠈⠃⠐⠁⠘⢇⠸⠐⠃⠘⠊⠩⠄⠠⣀⠀⠀⠈⠣⠴⠇⠸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣠⠞⣉⠑⠆⠀⠀⣠⠞⣤⠀⠀⠀⠀⠀⡀⣿⠀⣄⡶⢰⡆⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⣯⠸⣤⠀⠂⢰⠋⡸⢰⡇⣴⠋⡅⣾⠁⡗⡟⣠⠟⠇⢼⢁⣠⠏⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⣇⡀⠀⠀⠀⡠⠁⢹⠀⠻⠐⠃⡏⠉⢰⡇⠀⠄⢂⡟⣭⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⣠⠖⠒⠂⠠⢀⣀⠉⠓⠒⢩⠤⠄⠀⠀⠀⠀⠐⠇⠀⢘⡁⣠⠈⠁⠈⠷⢰⠆⠀⠺
⠀⠀⠀⠀⠀⠀⣿⢀⠀⣤⢠⡟⠑⠃⠀⠀⡈⠓⠲⠀⠀⠀⠀⡷⢰⢷⡟⢸⣇⢸⣃⠔⠀⠻⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠈⠁⣸⠁⣾⠁⡶⠻⢸⡇⣿⠀⠀⠀⠀⠀⠘⣇⣾⣈⣁⣀⠈⣀⠤⠖⠒⠋⢳⠄
⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⢻⠘⠧⠔⡀⠁⠉⠀⠀⠀⠀⠠⡟⣩⠋⢀⡠⠜⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⣴⠋⠙⠀⠀⠀⠀⠀⠀⠀⢀⠠⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠄⣀⣀⡀⠠⠄⠂

⠀⠀⠀⡀⠀⠀⡀⠀⠀⣀⠀⠀⠀⡀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⡄⠀⠀⢀



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢼⡏⠀⣿⢢⣽⣷⡀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢲⣾⠀⠀⣸⣿⡘⠷⣤⣿⡾⠁⣿⡇⢹⡏⠙⣦⡀⢷⣆⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⢸⣿⠀⠀⡿⢻⣧⠀⠀⣿⡄⠀⣿⠇⢸⡇⠀⣿⡇⠈⢿⡆⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣸⣿⠀⢰⣧⣼⣿⡄⠀⣿⣇⣰⠏⠀⢸⣧⡴⠟⠀⠀⠘⣿⡾⠁⠀⠀⠀⠀⠀⣠⠤⢤⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠛⠓⢦⠀⠀⠀⠀⣿⠟⠋⢹⣿⠀⣸⠁⠀⢸⣷⠀⣿⠟⠁⠀⠀⣸⣇⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⡞⠁⢀⡀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⢠⣿⠁⢰⣶⠀⢸⠃⠀⠀⠀⣿⡀⠀⠸⠿⠀⠋⠀⠀⠀⠀⠀⣿⠅⠀⠀⠀⠀⠀⠀⠀⣀⣀⡉⠉⠛⠷⣦⣀⠀⠀⠹⢤⠼⠇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣆⠀⠙⠛⠁⠀⣠⣴⠞⠋⠁⠀⠀⠀⣀⣀⣀⡀⠀⠀⠐⠛⠃⢀⣤⡶⠒⠒⡄⠀⡀⠈⠙⢷⡄⠀⠈⠙⢷⣤⣀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⢀⣤⡤⢬⡙⠷⢤⣤⠴⠟⠋⠁⠀⠀⣠⣴⡾⠿⠛⠛⠛⠛⠛⠛⠒⠶⣴⣿⣧⣀⣀⣠⣤⡴⠃⠀⠀⠈⣿⠀⠀⠀⣀⡈⠉⠙⠋⠉⠀⠀⣿⣷
⠀⠀⢀⣴⡿⠁⠀⢸⣿⠀⠀⠀⠀⣀⠀⠀⠀⣾⠟⠁⠀⠀⢠⡄⠀⠀⠀⠀⠀⣼⡿⠁⠀⠉⠉⠉⠉⠀⠀⠀⠀⣠⡟⠀⠀⣾⡏⠉⢧⠀⠀⠀⠀⠀⣿⡏
⠀⠀⣾⡿⠀⠀⠀⣿⡿⠀⢀⡖⠋⣿⣷⠀⠀⣇⠀⠀⠀⣠⣾⠇⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⢀⡀⠀⠀⠀⠐⠛⠉⠀⠀⠀⠙⠷⣄⣸⣀⣀⣀⠀⢸⣿⠁
⠀⢸⣿⡇⠀⠀⢸⣿⡇⣰⠏⠀⢠⣿⡟⠀⠀⠈⠓⠶⠟⠛⠁⠀⠀⣤⠀⠀⣾⣿⠀⠀⠀⣰⣿⠃⠀⢀⠀⠀⠀⢀⣀⣀⢠⣄⠀⠀⡽⠁⣸⣿⠂⣾⡏
⠀⢸⣿⡇⠀⠀⣿⡿⢰⠋⠀⠀⣸⣿⠃⠀⢀⡀⠀⠀⣠⡄⠀⠀⠀⠈⡇⠀⢿⣿⠀⢀⣼⣿⡟⢠⣾⠋⡽⠀⣴⡟⠁⣨⣿⡏⠀⢰⠇⢠⣿⡟⣀⣈
⠀⠈⢿⣇⠀⢰⣿⣧⡏⠀⠀⠀⣿⡟⢠⣾⠏⣹⠀⣾⠏⠀⣴⡗⠀⠀⡇⠀⠘⢿⣤⠞⣸⣿⠃⣿⡷⠊⢀⡼⣿⠁⡴⢻⣿⠀⢠⠏⠀⣾⣿⠙⠷⢸⣧
⠀⠀⠀⠉⠀⢸⣿⣿⠀⠀⠀⢸⣿⡇⣿⡿⠚⢁⢾⣿⠀⢠⣿⡇⠀⡼⠁⠀⠀⠀⠀⢀⣿⡏⠀⠹⣧⡴⠋⠀⠙⠚⠁⠘⠿⠴⠋⠀⠀⠻⣧⣀⣰⡿⠃
⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⢸⣿⡇⠙⠷⠖⠉⠀⠙⠓⠁⠈⠙⠊⠀⠐⠒⠒⠶⢦⣼⣿⠀⠀⠀⠀⠀⢀⣤⡶⠒⠒⢤⠀⠀⠀⢤⣄⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢿⣇⠀⠀⠀⢠⡤⠀⠀⣠⡴⠶⠾⠒⠶⠶⣦⣤⣼⣿⠉⠛⠶⢦⣤⡄⢸⡏⠀⢹⡆⠀⡇⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠸⣧⡀⠘⠁⠀⠀⠀⠀⠀⢀⣴⡿⠛⠻⢿⣶⣤⣄⣀⠀⠈⠓⠒⠛⣁⡼⠁⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠦⣤⣤⣤⣤⠴⠞⠛⠁⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠿⠿⠟⠋⠀⠀⠐⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠂⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐

⠀⠀⠀⠐⠷⣤⣀
⠀⠀⠴⣶⣟⡉⣠⠀⣶⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣦⡀⢰⡶⢦⡀
⠀⠀⠀⠀⠀⣰⠏⠀⣿⠶⣃⣀⠰⣆⣾⠀⠀⢀⡀⡀⣶⠲⠂⡀⣠⢀⡀⠘⢷⡟⣠⠶⡀⡿⠳⡌⢷⠾⣄
⠀⠀⠀⠀⠀⠉⠀⠀⠁⠀⣿⣼⠇⢸⡇⠀⠀⣼⣷⡇⣿⣉⡀⢿⣿⡞⠀⠀⠀⠷⠘⢿⡵⠆⠀⠀⠘⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⡄⠀⠀⠋⠘⠁⠀⢀⡀⢘⣈⠃⠀⠀⠀⠀⠀⠈⠀⣀⡠⢖⣂⣘⣷⣤⣤⡤⣤⡤⣄⡀
⠀⠀⠀⠀⠀⠀⣀⠢⠀⠀⠀⠀⠀⡸⠹⣦⡀⠀⠀⠀⠀⠥⣿⡿⠀⢉⠀⠀⣀⣠⣴⣾⣿⣿⣿⡿⠿⣷⣮⣍⢛⣿⣾⣿⣶⣿⠆
⠀⠀⢠⠴⣾⠿⠛⠟⠏⠀⠀⠀⢀⡇⠀⠈⣷⠀⠀⠀⠀⠀⠉⢀⣭⣿⣿⡿⢻⣟⢸⣿⣿⣫⡶⠶⢦⣄⣹⣿⠿⠞⣋⡭⠀⠀⠈
⠀⡆⠈⣘⡛⣖⡖⢶⠘⠀⢄⠀⠎⡇⠀⠀⢸⠃⠀⠀⠀⣤⣶⣿⢛⠻⣿⠃⢈⣿⣸⣿⣿⣿⢀⣄⠀⢛⣭⣶⣾⣿⣿⡥
⠀⠈⠛⠉⢩⢿⣱⣏⠃⠀⠄⠀⠘⠁⠀⠀⠸⠀⣠⣴⣿⣯⠸⣷⡬⢀⣿⣿⣾⣿⣿⠿⢿⠟⠚⠛⢀⡼⠃⠀⠀⠉⠁
⠀⠀⠐⠒⢼⡈⢿⣿⣇⠀⠀⠀⠀⠀⠄⠀⣡⣾⣟⣛⡻⢿⣦⣿⣷⠿⣻⠿⠛⠉⠀⠀⠀⠉⠀⠀⠉
⠀⢰⢺⣿⣶⣦⠛⢛⢝⣷⣤⡤⣤⠀⠀⢰⣿⣿⣿⣿⣿⡆⠙⣭⣾⣿⠎
⠀⠈⢿⢣⣨⣾⡦⠀⣾⣹⣏⠁⣦⠄⢠⠼⢹⣿⣿⠾⠛⡽⠀⠙⠋⠁
⠀⢸⠸⣿⣿⣿⣷⣬⣿⡉⢡⠠⠇⠀⠀⠀⠀⠘⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠉⠉⠁⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⡀⠠⢀⠀⠀⢱⠀⠀⣀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⣶⡆⠀⠀⣠⡟⠁⠀⠀⠘⠀⠀⠚⠀⢀⠎⢠⠏⠁⣀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢉⠵⡖⠢⢤⠀⢸⣇⠀⠀⢰⡿⠀⠀⣰⡟⠀⠀⠀⠀⠈⠢⠀⠠⠐⠉⠀⠸⢄⣀⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣴⠁⠐⠃⠒⠚⣀⣀⣭⣥⣤⣿⣇⡀⢰⡿⠁⢀⣀⡀⢀⡀⡀⢀⣀⣀⠀⡀⠀⣀
⠀⠀⠀⠀⢠⡎⠀⠓⠀⠀⢻⣟⣒⣒⣩⡭⡶⠟⠋⠁⣼⡏⠈⢻⣿⢃⡴⢋⣴⢃⣾⡃⣹⣾⠋⣸⢸⠇⣰⡟⠀⠀⠀⠔⠁
⠀⠀⠀⠀⠈⢳⣄⡀⠀⠀⠀⢈⣉⡭⠖⢋⡄⠀⠀⢰⡿⠀⠀⣼⡏⠸⠗⠋⠿⢸⠋⠛⢹⠃⠛⠃⠛⢋⡿⠀⠀⠈
⠀⠀⠀⢀⡀⠀⢈⡙⠛⠛⠛⠉⠁⠀⠀⣾⡇⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⢀⡟⠀⠀⠀⠀⣼⠁
⠀⡠⠚⠁⠀⠀⢀⠈⢳⡀⠀⠀⠀⠀⣸⡟⣷⠀⣼⢇⡴⣲⢠⡆⣴⠀⡠⠊⠘⣃⠀⠈⡁⠀⠀⠀⣰⠃⠀⠀⣸⡟
⡼⠁⠀⠀⠀⢇⠈⠀⡸⠁⠀⠀⠀⣰⡟⠀⢸⣰⡟⢿⣋⠔⢸⡗⣿⠞⠁⣠⠞⠁⠀⠀⠀⠀⢀⠴⠁⠀⠀⢠⡟
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢸⣿⠁⠈⠁⠀⠀⠀⠉⢀⣼⠏⠔⠹⠆⠀⠈⠈⠀⠀⠀⠀⢀⡞
⠹⣄⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⠀⠁⠀⠎⢡⠉⣦⠀⢀⣾⠏⡠⢤⢀⡀⡀⢠⡄⠀⠀⠀⠀⡜
⠀⠉⠛⠶⠦⠤⠄⠐⠈⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⣿⢀⣾⡟⣾⠕⢡⣏⣴⠃⠈⣽⠋⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⡀⠈⠳⡀⠀⠀⠉⣽⡟⠄⠙⠂⠉⠁⠛⠃⠀⢻⡀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠘⠃⠀⡇⠀⠀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠀⠀⠊⠀⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⠀⠀⠀⢀⡤⠚⠁
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠂⡄⠠⢲⠖⣠⡄⢄⢀⣀⠀⠀⠀⠀⠀⠐⠈⠀⠐⣂⡀⠀⠉⠁⠀⢀⣠⡄
⠀⠀⢀⠇⢹⣿⣶⡦⠄⢀⡶⢀⡀⢰⠀⠟⠘⠃⠸⠠⢘⠆⡟⢸⠃⡞⠹⡆⡔⢠⡄⢤⣦⣠⢧⣤⣄⣀⣠⣤⢠⡿⠁
⠀⠀⢸⣶⠉⠛⡏⠀⠀⠀⠈⣿⣤⣸⡽⠦⠤⣄⡀⠀⠀⠀⠀⠈⠈⠁⠂⠙⠰⡇⡆⢀⠿⢳⢼⣟⠙⠉⠀⢸⢸⣿
⠀⠀⠀⠈⠀⡸⠀⠀⠀⠀⠐⣿⡇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⢻⣄⡀⠀⠨⢸⡙
⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⢌⣿⠇⠀⠀⠀⠀⠐⢦⠀⡴⢱⢆⡇⡘⠀⢀⣤⠐⢄⠀⠀⠰⣗⠀⠀⠹⣿⡟⠁⣸⡧
⠀⢀⡠⠾⠁⠀⠀⢀⠀⠀⠈⣿⠀⠀⠀⠀⣀⣀⠘⠀⠁⠉⠈⠃⠄⠤⠏⣘⣰⠋⠀⠀⠐⣯⣀⣀⣀⠙⢆⠀⢼⠃
⣼⣿⣷⡶⠖⠀⠀⠀⠀⣀⡀⣈⢄⠀⠀⠘⡏⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠿⣿⠉⠀⠀⠻⢸⣤⡀
⠙⠿⡟⠀⢀⢄⡀⣤⠀⡵⢐⡿⠊⡖⠀⠀⡟⣼⢀⡯⠐⣆⣴⠰⠂⢀⡄⡀⠀⠀⠀⢀⠀⢠⡇⠀⠀⠀⠀⠀⠈⠸⠅
⠀⠀⠀⠐⢸⠾⠕⢉⣦⡥⣤⣲⡖⠁⠖⠀⠇⠒⠚⠳⠠⠭⢘⣁⠀⢈⡇⢰⠇⡀⡆⣸⡗⣾⡓⢦⣀⣠⣦⣀
⠀⠀⠀⠀⣼⡾⠷⠤⠈⠀⠀⠀⠀⣀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠈⠚⠀⠃⠃⠻⣇⡀⠀⠀⠉⠉⠉⣙⡿⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⢀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠒⠘⡠⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠃⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠘⠑⠆⠆⡰
⠀⠀⠀⠀⠀⠀⢎⠆⠀⢀⠠⠕⢄⡨⠀⠢⠔⠈⠥⡃⠀⠀⠀⠀⠀⠀⢀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠈⡜⠂⢄⡨⠠⡈⠠⢊⣃⠀⠀⠰⡱
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⡔⡠⠀⢄⠊⠂⢌⣈⣀⠊⠀⠀⠀⠀⠀⢀⠂⠀⠀⡀⠀⠀⠀⢠⠀⠀⠃⠀⠈⠂⠰⠄⡨⠠⡐⢆⠁⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠈⠉⠀⠀⢀⠀⠀⠀⠁⡄⣐⠁⠎⠉⠀⡐⢆⠁⠀⠀⠀⠀⡠⠒⠀⠀⠀⠐⠄⠀⠈⠁⠀⠀⠘⠠⢈⢰⠁⠂⠑⠀⢄⠉⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠐⠉⢑⠀⠀⠀⠀⠢⠘⠬⠀⠒⠀⠁⠀⠀⣀⠀⠀⢃⣀⣀⣀⣀⣀⣠⠀⠀⣀⠀⠀⠀⠑⠉⠼⠀⡀⠣⡉⠉⠀⠀⠀⡊⠉⠂
⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣠⠑⠈⢀⠀⠀⠀⠀⠀⣀⠁⠊⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⡠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠀⠀⠀⠀⠀⠀⠡⠀⠔⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⠀⠀⠀⢀⡀
⠀⠣⠔⠀⠀⠀⠑⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠊⠀⠀⠀⠢⠐
⠀⠀⠀⠀⡄⠀⠀⠀⠀⡄⠀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠈⠂⠁⠀⠀⠀⠀⠀⠊⠀⠀⠀⠲⢤⢧⠴⠂⠀⠀⠀⢠⡀⠀⠀⠀⣠
⠀⢀⠄⠠⠇⢨⠎⢦⠄⠧⠀⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠑⠠⢀⡀⠀⠀⠀⢀⡀⠄⠂⠁⠀⠀⠀⠀⠲⠋⡏⠑⠂⠀⠠⠀⠼⠀⡴⠱⡅⠸⠄⠠⡀
⠀⠘⡤⠤⠤⢠⠄⢤⠤⠠⠄⡄⠀⠀⠀⠀⢠⣄⡹⡴⠃⠀⠀⠀⠀⠘⠀⠁⠀⠃⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢠⠤⠄⠤⡤⠠⡤⠤⠠⢤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⣎⠙⠃⠀⠀⠀⠄⠀⠀⠀⢠⠀⡐⠐⢐
⠀⠀⠀⠀⠀⠰⠀⠠⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠤⠀⠆
⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⣇⠄⠀⠀⠀⠀⠀⠀⠰⠉⠆
⠀⢆⠀⡰⠀⠀⠀⠀⠀⠀⠀⡀⠌⠐⠰⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣘⠒⠃⠂⠃⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠲⡀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠁⣆⠄⠤⠀⠀⡃⠈⡀⡀⠀⠀⠎⢀⠈⢄⠉⠀⠈⠙⠀⠄⠈⠁⢂⠈⢉⠆⠀⠀⠀⠀⣘⢀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡀⠀⠎⠀⠋⠁⠸⠀⠐⠈⣠⠀⠒⢀⠴⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢰⠁⢈⠄
⠀⠀⠀⠀⠀⠀⠀⠠⠠⠔⢢⢀⡀⠀⠀⠀⠀⠧⠠⠀⠠⠠⠦⠊⠉⠱⠤⣆⠘⠁⠠⠠⠍⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠂⡄⠄⠀⠀⠀⠁⠁
⠀⠀⠀⠠⠤⠀⡆⠐⢢⠀⠠⠀⠁⠀⠄⠀⠀⠀⠀⠠⠠⡀⠀⠄⢤⠀⠀⠀⣀⢀⠀⠀⢀⠀⢠⢄⠀⠀⠀⠀⠀⢀⠀⠈⠀⢀⠀⡄⠈⠠⠀⠦
⠀⠀⠀⠔⠀⡀⠒⢈⢌⠀⣨⠐⡤⠀⠘⢰⠀⠀⠀⠁⠀⠈⢠⠀⢸⠠⠤⠤⠆⠀⢄⠁⠀⣀⠂⢀⠂⠀⠀⠀⢆⠊⠀⢄⠀⣉⠀⡠⡅⠒⠁⡀⠰⡀
⠀⠀⠀⡒⠁⡌⡠⡀⢁⢄⠉⣀⠠⠖⢄⣱⠀⠀⠀⠀⠀⠲⡀⠀⢘⠀⠄⠠⢀⠀⠈⢀⠆⠈⠀⠈⠀⠀⠀⠀⢎⡠⠀⠄⣀⠈⠠⠈⢀⢀⠉⠈⠐⠁
⠀⠀⢨⡈⠂⠃⠀⣠⡈⢈⠀⠑⠡⣀⠄⠀⠀⠀⠀⠀⠐⠁⠈⠂⠲⠀⠀⠀⠐⠑⠀⠂⠐⠀⠐⠀⠀⠈⠀⠀⠠⠠⢀⡄⠙⢀⡀⢀⢄⠀⠘⠐⠁⡄
⠀⠀⠘⠠⠀⠊⠑⠭⡀⢈⡋⠔⠄⠀⠐⡀⠀⠀⠀⠤⠐⠄⠠⠐⡖⠒⠀⢐⠀⡠⠀⠄⠐⠀⢀⠂⢄⠀⠀⠀⠈⠂⠀⢠⠢⠜⣀⠀⡩⠜⠁⠀⡄
⠀⠀⠀⠀⠑⠁⡄⠀⠀⠐⠢⠀⠣⠌⠁⠀⠀⠀⠀⠈⢂⠈⠁⠼⠀⠙⠋⣃⠠⢀⢆⠈⡆⠀⠃⠀⠀⠀⠀⠀⠀⠁⠱⠬⠀⡀⠊⠀⠑⢢⠀⠒
⠀⠠⠂⠲⠀⠀⠈⠈⠀⠉⠈⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠘⡇⡠⠠⡀⠐⡀⣆⠈⠠⠀⠀⢠⡄⠀⠀⠀⠀⠁⠉⠂⠋⠁⠀⠀⡎⠉⠂
⠀⠈⠂⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠈⠁⠉⠉⠁⠀⠀⠐⠂⠉⠀⠁⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣷⠈⣟⢠⡄⣿⡄⢀⠀⣤⢀⣠⡄⠀⢠⢠⡀⣤⣤⣄⣄⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠙⠃⠙⠈⠿⠛⠁⠙⠛⠃⢨⢹⡇⠀⠘⢹⡏⠁⠈⠙⠛⠀⠀⠀⢙⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣶⠞⠛⠳⢶⣤⡀⠀⠀⠀⣀⠀⠀⠀⠀⣀⣴⣶⡿⠿⠷⢶⣤⡀⠀⠀⠀⣴⣾⠟⠛⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣾⡟⠙⢷⣄⠀⠀⠹⣿⣆⠀⠀⠉⠀⠀⠀⣴⣿⡟⠁⠀⠀⠀⠀⠈⢿⣦⠀⢸⣿⠁⠀⠀⠀⠘⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠹⣆⠀⠀⢿⣿⡄⠀⠀⠀⠀⣀⣿⣏⣤⣄⡀⠀⠀⠀⠀⠸⣿⣧⠸⣿⡀⢀⣤⡀⠀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⢠⣿⠀⠀⣾⣿⡇⠀⢀⣤⣾⠟⠻⣧⡀⢈⣿⠀⠀⠀⠀⠀⣿⣿⡆⠙⠷⣄⣼⠇⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣶⡿⠃⠀⢀⣿⣿⡇⢠⣾⣿⠏⠀⠀⢀⣽⠫⣅⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⣸⣿⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⢀⣿⣿⡇⠀⠀⠀⠘⠓⠀⣿⣧⠀⠀⢀⣿⣿⣿⠀⠀⠀⠀⠀⣰⣿⣷⠏
⠀⠀⠀⠀⠀⠀⠀⡷⠆⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠃⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⣾⣿⣿⠇⠀⠀⠀⠀⣰⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⣀⣼⣿⣿⠃⠀⢸⣿⣿⠀⠀⠀⠀⠀⢀⣾⣿⠃⢀⣾⣿⣿⢋⣠⣤⣤⣤⣼⣿⡿⠁⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠿⠛⠛⢛⣿⣿⣿⣷⣤⡀⠀⠻⣿⣆⠀⠀⢀⣠⣿⡿⠃⢠⣾⣿⡿⢃⡾⠋⠁⣩⣿⠟⠻⢿⣦⡀
⠀⠀⠀⠀⠀⠀⢸⡿⠉⠀⠀⠀⣠⣾⡿⠋⠀⠈⠻⣿⣦⠀⠉⠛⠿⠿⠟⠛⠁⠀⣴⣿⣿⠟⠀⠈⠻⠶⠛⠋⠁⠀⠀⠀⠙⢿⣆
⠀⠀⠀⠀⠀⠀⠘⢧⣀⣠⣴⡾⠟⠁⠀⠀⠀⠀⠀⠀⠛⣠⣴⣾⣿⣿⠿⢿⣿⣿⣿⡿⠃⠀⠀⢀⣠⣶⡿⠟⠛⠛⠷⣄⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⡄⠀⠀⠀⣰⣿⠟⠛⠉⠀⠀⣠⣾⣿⡿⠛⠻⢦⣄⣴⣿⠟⠁⠀⠀⠀⠀⠀⣹⡇⠀⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⢻⡀⠀⠰⣿⠁⠀⠀⢀⣤⣾⡿⠟⠉⠀⠀⢀⣴⡾⠟⠓⠦⠤⣄⣠⣤⣤⠾⠛⠁⠀⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢀⡀⠀⠹⣄⠀⠙⠓⠶⠾⠟⠋⠁⠀⠀⠀⣠⣴⡿⠋⠁⠀⣠⡄⠀⠀⠀⣀⠀⠀⠀⠀⠀⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⣱⡞⠁⠀⠀⠈⠙⠢⣄⣀⣀⣀⣀⣠⣤⡶⠟⠋⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠙⣿⢡⡤⣀⡀⠀⠀⠀⠉⢉⠉⠁⠀⠀⠀⠀⠀⠀⣀⢠⡄⠀⠀⣠⡐⢶⡄⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢁⡿⠳⡟⣤⠀⠀⣿⠀⣿⢀⠀⡀⠀⡀⠀⠹⢿⡇⣖⠘⠋⠛⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⢸⡏⠀⠀⠟⠀⠟⠟⠈⠷⠻⠇⠀⠀⠈⠃





⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡟⠁⠉⠉⡉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣀⣠⡄⠒⢯⣅⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣻⠃⢙⡗⠰⣧⡀⠈⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠙⢿⠀⠀⠽⢵⣍⠓⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢳⡀⠈⠣⣀⠀⠉⠘⢻⡏
⠀⠀⠀⣀⠤⣤⣄⡀⣀⢤⢤⢤⣀⠀⠀⠀⢀⡜⢀⠝⠀⠘⡶⠾⠟⣲⣶⡟⠀⠀⠀⠀⠀⡤⢂⣶⠶⠦
⠀⠀⡼⠁⠸⣿⣿⣿⣧⣤⣳⢣⢫⣆⠀⠀⡾⠐⠧⠤⢀⠀⠀⠀⠘⠛⢍⡹⠀⠀⠀⠀⠸⣷⣾⣷⣷⣄⣦⣄⡀
⠀⢰⡇⠀⠀⠙⢿⡿⣸⣇⣹⣿⠽⡿⠞⠛⡞⠩⡑⠲⡅⠀⠀⣠⡖⠒⠉⠀⠀⠀⠀⠀⢤⡉⠋⠙⠛⠿⠋⣙⣛⡆
⠀⠀⢷⡀⠀⠀⠀⢠⠟⠉⢋⡤⠊⠀⠀⠀⠙⢄⣀⣀⣤⠐⠈⣽⠁⠀⠀⠀⠀⠀⠀⠲⣧⢉⠀⠀⢀⣀⠀⢹⣨⣿
⠀⠀⠀⠙⠒⠒⠚⠁⠉⠉⠏⠀⠀⣠⣤⣴⡶⣲⡿⣿⡟⢀⣼⢿⣦⣀⠀⠀⠀⠀⡎⢀⠔⠃⠀⢰⣶⡌⢁⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁⠁⣠⣘⣣⣯⡭⠼⣿⣿⣿⡿⠂⢸⠁⡶⢤⡀⠀⠀⠀⢠⣾⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⠀⢫⣤⣬⣴⣾⢹⣿⠏⣠⣶⠗⢏⠈⠛⠂⠁⠀⠀⣠⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠀⠀⣠⣿⣿⣿⣿⡿⡘⠛⠀⠻⠷⣶⣮⡍⢙⠟⠛⠋⣱⣿⠏⠉⠛⢥⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⡿⣫⠺⡇⠀⠠⠤⢄⣉⣉⣠⠋⠀⠀⣼⣇⣰⣍⢳⣾⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⡇⢐⣀⠂⡀⠀⠀⠀⠈⠉⠀⠀⠀⠀⣿⠙⠛⠻⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⢦⣤⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡄⠠⠤⠤⣶⣶⣶⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⡇⠀⠀⠀⢻⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠘⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⢻⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠸⣿⣿⣿⣿⣿
⠀⠀⠀⢠⣄⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢿⣿⣿⣿⣿⡇
⠀⠀⢠⣿⣿⣷⣄⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠈⣿⣿⣿⣿⣿
⠀⠀⢿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⢹⣿⣿⣿⣿⣧
⠀⠀⠀⠙⢿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⢿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⣄⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠘⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣦⡀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠸⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣷⡄⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⢿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣄⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⡟⢣⣀⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣽⣿⣿⣆⠀⠀⠘⣿⣿⣿⣿⣿⣿⡿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡆⠀⠀⢹⣿⣿⣿⣿⣿⡇⠀⢸⠟⠛⠉⠁⠀⣤⣾⣿⣿⣯⡍⢉⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡄⠀⣬⣿⣿⣿⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣷⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⠀⠀⠀⢸⣿⣿⣿⣧⣾⣿⣿⠟⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⣩⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣭⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣦⠁⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣟⠁⢹⣿⣿⠇⠀⠀⢀⣤⣴⣶⣿⡿⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣀⣘⣿⡏⠀⠀⠀⡿⠟⠛⠉
⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⣰⠛⡿⠋⣿⣶⠀⠀⡄⠰⠀⢾⠿⠿⠿⠛⠛⠉⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡀
⠀⢿⣤⣈⣀⠀⠱⡶⠁⠈⠃⠀⠀⠀⠀⠀⠀⠀⣠⣿⣷⠀⠀⠀⢠⣴⣆⠐⠀⢀⠠⠄
⠀⠈⣿⣿⡇⠀⠤⠀⠀⠀⣠⣤⣶⣤⠀⠀⠀⢀⣾⣿⣏⠀⠀⠀⠀⣦⡹⡆
⠀⣴⣾⣿⣇⠀⠀⠀⠀⠀⠸⣇⠀⠈⣄⠀⠀⠀⣿⣿⣿⣄⠀⠀⠀⣍⡻⢷⡄⠰
⠀⠸⣽⣿⣿⣇⢰⠀⠀⠀⠀⠘⣄⠻⣾⣷⣶⣦⣿⣿⡋⠸⡧⠀⠀⢹⣷⣦⣴⠆
⠀⠀⠀⡾⠁⢡⣜⣀⡄⢀⡀⠀⠘⢧⠈⣭⣿⣿⣿⣿⣿⣆⠀⢹⠀⠺⠁⡿⠗⣤
⠀⠀⠸⡆⠀⡴⠉⠁⣠⠞⠁⠀⠀⠈⣇⠸⣿⡻⣿⣿⣿⣿⣧⣾⣧⢰⠄⣇⢺⡄
⠀⠀⠀⠘⠒⡿⠚⠋⠁⣀⣠⣶⣤⣤⣯⣴⡎⢻⡝⢿⣿⣿⣿⣿⡿⠁⠘⣷⣄⡀
⠀⠀⠀⠀⠀⢡⡀⣰⡿⠋⠁⠀⣀⣹⡟⢉⣻⣧⣽⣿⣛⠛⣹⡟⠁⢰⣦⡿⣯⣁
⠀⠀⠀⠀⠀⠀⠣⣘⠀⠀⠐⠀⣹⠟⢠⣾⠿⢋⣹⣾⣧⣠⠏⠀⠀⠀⢛⡷⣿⠸
⠀⠀⠀⠀⠀⠀⠀⡿⠁⠓⠚⠻⠃⣤⣾⠉⣠⡾⣿⣿⣿⡇⠀⠀⠀⠘⣯⣧⡉⡀
⠀⠀⠀⠀⠀⠀⢀⣶⣦⡀⢤⡀⠀⠹⣇⠀⣏⡀⢈⣿⠿⣷⠀⠀⠀⠐⣿⠻⠇
⠀⠀⠀⠀⠀⠀⠈⢉⣹⠃⠀⢹⠀⢀⠛⠀⢹⣇⠻⣿⡳⣾⠃⠀⠀⠀⠗
⠀⠀⠀⠀⠀⠀⠀⠈⠹⠿⢦⠍⠀⢺⠁⠀⠀⢹⠐⢿⣵⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠓⠲⠲⢆⣀⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠈⠀⠰⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠀⠀⠀⢀⠀⠀⠶⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣀⣠⠀⠀⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠲⣿⣡⣿⣯⠛⠁⠀⢸⡹⠋⠳⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣟⡙⢣⣀⣽⣿⣿⡀⢀⣽⣍⠀⠀⠘⠷
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣟⠻⠛⠿⠹⢿⣷⣾⣿⣟⣡⣄⣷⣶⣷⣄⠀⠀⠀⠀⢀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠤⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠠⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⢠⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⡈⠁⠀⠀⡰⠁⠀⢲⠃
⠀⠀⠀⠀⣀⠀⠀⢀⠀⠤⠒⠉⠉⠀⠀⠀⠀⠀⠀⠁⣄⠓⠢⠤⣀⠀⠀⠀⠀⠀⠀⡜⠁⠉⢢⠀⢰⠁⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢠⠏
⠀⠀⠀⠀⠁⠀⠀⠈⢖⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠉⠒⠄⢀⠀⠀⠑⠀⠀⢸⡆⡇⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠃⠀⠀⠀⠀⠀⢀⡦⠀⠀⠀⠀⠀⠀⢸⢳⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⡆⢰⡇⢀⠄⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⣀⠀⠀⠀⡌⢸⠀⠀⠀⠀⠀⢰⡀⡜⠩⣿⠴⠋⠓⢩⡇⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⡠⠊⠀⠀⢀⠎⢠⠀⢀⠜⠀⠸⡀⠀⠀⠈⠈⠀⢨⠃⠀⠁⠀⡠⠊⡰⠀⠀⠀⠀⢼
⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠑⢄⠀⠀⢀⠴⠅⠀⠀⢀⡠⠊⢤⡀⠀⢀⠆⠀⠈⠈⠁⠀⠀⠀⠑⠒⠀⠌⠀⠀⠀⠀⠀⠀⠀⢑⠚⣄⣀⠄⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠑⠁⠀⠀⠀⠀⠀⠈⡏⡇⠀⠸⡑⠈⠁⠀⠀⢰⡇⠀⠸⠀⠀⠀⠀⠀⠀⢀⣀⠠⠀⠘⠀⠀⣰⠂⢀⢄⠇⡀⣹⣀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠇⢀⠃⠁⠀⠀⠀⠀⠀⡇⠀⡇⠀⣀⡄⢀⠁⠀⡸⠀⠀⠀⠘⠒⠊⣹⠠⠚⠁⠊⠙⠁⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠈⠀⠀⠀⠀⠀⠀⢸⠹⡀⡇⢰⠕⠁⡞⡰⠥⠃⠀⠀⠀⡠⠔⢈⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠠⠀⢠⠃⠀⢳⡇⠠⠤⠊⠈⠀⣀⡀⢀⠐⠁⠖⢀⠂⠐⠒⠒⠚⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠐⠁⠀⠀⠈⠇⢀⡠⠔⠚⠉⠁⠀⠀⠀⠀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠄⠐⠀⢀⠎⠀⠀⠀⠂⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠂⠄⠐⠊⠁

⠀⢄⣀⡄

⠀⢠⠤⡀⢠⠈⠃⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⡀⣼⡀⠀⣼⠀⢰⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⢹⠀⡠⠚⣆
⡇⠸⠇⠀⢿⡄⠘⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⡘⠆⠁⣠⠋
⠃⠀⠈⠀⢀⣀⣀⣀⠛⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣴⠀⠘⠛⠀⣼⠁⠀⢀
⠀⢃⠀⠀⠀⠀⠀⠈⠛⢀⣤⢠⡇⠀⠀⠀⠀⠀⠀⢀⣀⢀⠀⠉⠘⠂⠀⠀⣀⣀⡀⢀⡀⠀⠀⠀⠀⠀⠘⠧⣄⠀⠹⠀⠀⠀⠀⠀⠠⠄
⠀⠈⢦⡀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡌⠐⠶⠀⠀⠛⠀⠉⢈⡵⠈⠹⡆⠀⠀⠀⠀⠀⠀⠈⡇⠀⠳⣤⡀⠀⢀⡤⣤⠁
⠀⠀⠀⠉⠲⣄⠀⢾⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠄⣀⣀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⡼⠃⠀⠈
⠀⠀⠀⠀⠀⠈⡀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠉⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠼⠛⠀⠀⠀⢀⡛⠀⢠⡤⠚⠁
⠀⠀⠀⠀⠀⠀⣗⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⣹⠶⠒⠋⠉⠉⠉⠉⠁⠒⠲⣤⡀⠀⠀⠀⠀⠲⡅⠀⠀⠀⠀⣴⠋⠀⢀⡎
⠀⠀⠀⠀⠀⠀⢹⡄⠀⢻⡀⠀⠀⠀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠹⠆⠀⢀⡴⠁⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠈⢃⠀⠘⢣⠀⢠⠞⠁⠀⠀⠀⠛⠃⠀⠀⠀⠀⠉⠓⠀⠀⠀⠀⠀⠹⡀⠀⠀⡀⠀⢀⡞⠁⠀⢀⡾
⠀⠀⠀⠀⠀⠀⢀⣼⠦⠶⠛⢻⠇⠀⠀⠀⠀⠀⠀⡠⠒⠲⡄⠀⠀⠀⢀⣠⠾⠀⠀⠀⢫⡀⠀⢹⣦⠛⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠘⢇⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠋⢁⣼⠀⠀⠀⠘⣇⡀⢸⡟⠀⠀⢀⠾
⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢰⠀⠀⠀⠀⢸⣿⡗⠙⠛⠛⠛⠉⠀⢀⣀⣴⣿⡟⠀⠀⠀⠀⠀⠙⡿⢦⣀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠘⠀⠀⠀⠀⠸⣿⣷⣤⣤⣤⣤⡶⠛⠋⠉⢹⡟⠁⠀⠀⠀⢷⡤⠛⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⣷⠀⠀⠀⠀⠹⣿⣿⣿⠿⠟⠐⠒⠒⣴⠟⠁⠀⠀⠀⣠⠟⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠈⣆⠀⠀⠀⠀⠈⠻⠇⣀⣀⡠⠶⠞⠁⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⠟⠁⠀⠀⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠈⠙⠶⢦⣤⣤⣤⣤⣤⣤⡴⠶⠛⠉⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠳⠾⠙⠛⠛⠛⠋⠁⠰⠶⠶⠦⢄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠁⠈⠙⠛⠛⠛⠲⠦⢤⣤⡤⠤⠾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠐⠛⠋⣉⣁⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠘⠳⣤⡤⠔⠛⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢦⡤⠴⠶⢶⣶⠶⠖⠛⣿⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣀⡸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠⠀⠀⠀⠈⠉⠉⣿⠉⠙⠛⠟⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠧⠤⣼⡁⠀⠀⠀⠀⠀⠀⡰⠓⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⢘⡇⠀⠀⠀⢀⡤⠚⠀⠀⠀⢠⡇
⠀⠀⣀⠀⡀⣀⣠⡴⠖⠉⠀⠀⠀⠀⡀⠘⣧⣀⠰⠊⠉⠀⠀⠀⠀⠠⠺⡇
⠀⢘⠋⠤⠀⣀⣈⣡⡀⣀⡀⠀⢀⡠⠚⠛⠁⠉⢠⡀⠀⠀⠀⠀⡜⠁⣀⠃
⠀⠈⠑⠂⢤⣤⣤⣤⣤⣤⠀⠠⣏⠑⠂⠠⠀⠀⠀⣴⠄⢤⣤⠘⠓⢠⠌⠁
⣤⣤⣀⣤⣤⣤⣤⣤⣤⣠⣤⣤⣄⣉⣤⣤⣶⣶⣶⣤⣶⣤⣤⣌⣁⣠⣤⣄⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣤⣤⣠⣤⣤⣤⣄⣠
⣿⣿⣿⣿⠿⠛⡿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣀⣀⣀⣀⣀⣀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠒⠲⢶⣄⠀⠀⠀⣠⡴⠖⠛⠓⠶⣄⠀⠀⠀⢀⣤⠶⠒⠒⠶⣤⡀⠀⠀⢀⣴⠶⠒⠲⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⣠⠞⠛⠳⡆⠘⣇⠀⢰⡏⢠⡞⠛⠛⣦⠈⣷⠀⢀⡾⢁⡴⠛⠛⣦⠀⢿⠀⢠⡟⢀⡶⠛⠳⣦⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠀⠀⢠⡟⢠⡏⠀⢸⡇⢸⠀⠀⠀⣿⠀⣿⠀⠈⠛⠛⠁⠀⠀⣾⠀⣼⠀⠈⠉⠉⠁⠀⢀⣿⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⢠⡞⠁⠀⢸⡇⢸⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⢀⡾⠃⣴⠃⠀⠀⠀⠀⢸⡟⠋⢵⣞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⣴⠏⠀⠀⠀⢸⡇⢸⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⣰⠏⢠⡞⠁⠀⠀⠀⠀⠀⠈⠉⠛⢶⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⢠⡾⠁⠀⠀⠀⠀⢸⡇⢸⡄⠀⠀⣿⠀⣿⠀⠀⢀⡾⠁⣴⠏⠀⠀⠀⠀⢰⡶⢶⡀⠀⠀⣸⠃⣿
⣀⣀⣀⣀⣀⣀⣀⣀⣸⠃⠀⠛⠛⠛⠛⠛⣷⣀⣈⣷⡈⠛⠓⠛⢁⣰⣏⣀⣠⡟⠀⠘⠛⠛⠛⠛⢻⣆⣈⣷⣈⠛⠓⠛⠋⣠⣟⣀⣀⣀⣀⣀⣀⣀⣀
⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⣄⢀⡤⢠⢾⠀⠀⠀⠀⢠⡸⠇⢋⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡘⠘⡡⠀⠀⠀⠀⢸⡇⢸⡜⢸⡎⡰⠀⠀⠐⣒⢀⠂⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣢⠰⡈⠀⡀⠀⠣⠜⠣⠚⠓⠉⠉⠁⠀⠠⠈⡠⠃⡆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠁⢱⠀⠀⡄⣠⢀⣶⢰⢦⢰⢢⣄⢰⠀⡐⠁⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢣⠀⡧⡆⣼⣼⢸⠋⢸⠁⢸⠇⡜⠀⠀⠀⠠⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⠀⠚⠀⣀⠁⠇⠛⠃⠘⠘⠀⠘⠀⠈⠀⢰⡳⠀⠛⠀⢀⣤⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⡠⣾⠏⠀⢀⡀⢠⣞⠇⠀⣼⠇⡴⡏⣸⠷⠤⠴⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠖⠊⢀⡏⣠⠞⣽⠁⣾⠋⣠⠎⢾⠞⠀⠗⠁⣶⡷⠀⠀⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠾⠇⢠⠀⠈⠛⠁⠀⠓⠋⢉⠋⠀⣠⡀⠀⣷⣀⡀⠀⠀⠶⣦⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣠⠤⠖⢳⡄⢰⡿⢃⣾⢹⣇⡰⠃⢸⠁⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢸⡷⠛⠶⠋⠉⠀⣉⢀⡀⠸⣤⣠⡴⠋⢠⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⢖⡄⢀⢾⡇⢀⠞⡶⢹⡎⢸⡇⢹⠀⠀⠀⠀⠘⠆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠾⠁⠈⢐⡇⣮⠇⡽⠁⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣂⣀⣤⣌⣀⡀⠀⠐⠒⢭⡓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⢛⣫⣭⣭⣭⣭⣇⣀⣀⣀⣐⣿⣾⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢟⡵⡺⠟⠟⠛⠉⠀⠉⠁⠀⠀⠀⠀⠉⠹⣟⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢁⠏⠀⠀⠀⠀⠀⠈⠉⠙⠂⠀⠀⠀⠀⠀⠀⢿⡆⢘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠉⢹⠆⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⢆⠀⠀⠀⠀⠀⠀⠀⠀⢹⡍⠀⠀⠀⢸⠱⠀⠈⠔⠀⢀⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠸⠙⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢠⠃⠀⠀⠀⠈⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⡠⠖⢣⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠈⢦⢀⣀⢄⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢤⡆⠀⠀⠀⠀⠀⢴⠒⠚⠉⢀⡠⡾⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⢀⢀⠆⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠈⢶⠒⠚⣉⢴⠇⠀⠀⠀⠀⢀⡾⠁⠀⠀⣀⠔⡡⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠙⠲⠒⠁⠀⠀⠀⢀⣴⠏⠀⢀⡤⣄⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⣀⣀⠀⠀⢀⣀⣠⠴⡟⠴⠒⠊⣁⡠⠟
⠀⠀⠀⠀⠀⠀⡠⠤⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠈⢉⡩⠝⠋⢅⣀⡠⠃⡖⠂⠉⠁
⠀⠀⠀⠀⠠⡄⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠠⠤⡏⠀⠉⢉⣠⢴⠆⠀⠈⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠞⠁⠒⠒⠒⠒⠉⠑⠒⠉⠉⢀⡞⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⠤⠬⣉⠑⠂⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡀⠀⠀⠉⠓⠢⠤⣈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠺⣆⠀⠀⠀⠀⠁⠒⠤⣄⣀⡀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⣯⠙⡖⠒⠒⠃⠤⠤⠔
⠀⠠⢄⠀⠀⠀⠀⠀⢀⡔⢉⡴⠊⠉⠹⠦⠖⠒⠒⠓⠓⠂⢄⡈⠉⠉⠛⠒⠒⠒⣃⣀⠀⠀⠀⢀⠏⣸⠁
⠠⡑⢄⠀⠒⠤⣀⢠⠎⢨⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢇⠀⠀⠀⢀⡏⣀⠠⠔⣚⠖⠁
⠀⠘⢄⠳⣀⠀⠀⠉⠲⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣄⣀⡼⠒⠒⠒⠉⠁
⠀⠀⠀⠙⠢⡑⢤⡀⡸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡄⠀⠀⢹⡀⠀⢠⠿⡂⢸⠀⠀⠠⠒⠀⠉⣰⠃⣀⣬⢧⡀
⠀⠀⠀⠀⠀⢀⣴⣶⡇⣰⣋⣥⣼⣽⠁⣠⣷⡀⢧⠀⠸⢠⣵⠸⠀⠀⢀⣀⣤⣄⠁⠀⠛⠁⣠⠗
⠀⠀⠀⢠⣴⣾⢿⡗⢂⠏⠙⠋⠉⣏⡔⠁⠇⢧⠈⡌⠻⠇⠉⠀⢰⣾⠿⠛⠛⣋⡀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠠⠒⠓⡏⢀⠀⠀⠀⣧⣴⣶⣶⣾⡄⠀⠀⠀⠀⢺⣿⠀⣠⠄⠸⡅⠔⢄⠻⠤⠤⣄
⠀⠀⠀⠙⢠⣷⣶⣾⠀⠀⠑⠀⣴⡯⢋⡀⢀⠞⡃⣀⠀⠀⠀⠈⡿⣧⠀⠚⢀⣌⣢⠈⣆⠀⠀⡸
⠀⠀⠀⢀⣿⣿⣿⡏⠀⠀⠀⠘⣯⠀⠋⠒⠀⠳⡀⡈⢆⠀⠀⠀⠣⣀⡀⢰⠀⣈⣵⢀⣹⠀⠐⠁
⠀⠀⠀⢸⣿⣿⣿⠇⠀⣖⠀⠀⣞⠃⠀⠠⢠⣔⣤⡷⢸⠀⠀⠀⠀⢀⣱⠄⡁⣉⣋⣼⠟⣾⣿⣿⡟
⠀⠀⠀⣿⣿⣿⣿⣀⡄⠀⠀⠀⠀⠑⠒⣜⠄⠉⠛⠁⣼⣠⣶⣷⣴⣿⣿⣷⡀⣍⣄⣀⣶⣹⣿⣿⣷
⠀⠀⠀⢿⣿⣿⣿⣿⣷⣦⣤⣴⣶⣀⣀⣨⣍⢓⠓⠛⣽⣿⣿⣿⣿⣿⣿⣿⣷⡹⣿⠟⠛⢿⣿⣿⣻
⠀⠀⠀⠀⠙⠻⠿⣿⣛⣉⡉⠻⣿⣿⣿⣿⣿⣆⢻⣾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡉⠀⠀⠣⠭⠭⣍⡀
⠀⠀⠀⢀⣄⠀⣼⢋⠉⠈⠙⣰⠟⠈⢝⢻⣿⣿⣆⠃⢠⡀⠻⠟⢛⠛⠿⣿⣿⡿⠷⠀⠙⠀⠈⠉⢢⠇
⠀⠀⠀⠈⢁⡴⠋⠀⠀⠀⠈⢁⡀⠠⠒⢣⡿⣿⣿⣆⠀⠀⣀⣀⣉⣀⣶⣀⠀⢀⣀⠀⡤⢀⡠⢞⠉
⠀⠀⠀⠀⢼⠀⠀⠀⠀⠀⣴⡿⠂⠀⠰⠟⠑⣤⠻⣿⣆⠀⠈⠑⢯⡰⠎⣉⡼⠋⠀⢀⡴⠋⠔⠁⡄
⠀⠀⠀⠀⠘⣧⡀⠀⠀⠘⢉⠒⢦⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⡌⣮⣩⡇⠀⠀⣼⠋⠀⢄⡀
⠀⠀⠀⢠⠀⠸⣷⡄⠀⠀⠈⠣⣤⡤⠀⠀⠀⠉⢡⠀⠀⠀⠀⠀⠠⢁⡾⠏⠀⢀⠀⣏⠳⢄⡀⠈⠑
⠀⠀⠀⠈⢿⢦⡘⣿⣄⠀⠀⠀⠀⠑⣦⠀⡴⠞⠋⠀⠀⠀⠀⡠⠱⠃⠀⡸⠀⠉⠁⠈⢑⠦⡉⢦
⠀⠀⠀⢀⣈⣆⠁⠘⢿⣦⠀⢰⡀⠀⣼⡇⡀⠀⡠⠖⠉⠁⠀⣇⠀⣠⠏⠀⠀⠀⠀⠀⠈⠑⠬⣼
⠀⠀⠀⠈⠛⠟⣰⡾⠉⢿⣧⣸⡙⠛⢻⡇⠀⡜⠀⡠⠔⠒⢦⣜⡄⠱⠀⠀⠀⠀⠑⢄⠀⠀⣰⠿⠓
⠀⠀⠀⠀⠀⢰⣿⢀⣤⡄⠻⠿⠙⣛⠛⠁⠀⣿⠏⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠶⠀⠈⣢⡾⢣⠂
⠀⠀⠀⠀⠶⠀⠻⣿⣧⣧⣾⠿⣤⠿⠀⠀⢸⣿⠓⢄⡀⠢⠀⠀⠈⠳⠀⠀⣀⣠⠴⠚⠡⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⠀⠀⠀⠀⠀⠈⠉⠀⠀⠈⠉⠛⠒⠚⠛⠩⠭⠁⠒⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⣤⣼⣿⣿⣿⣷⣾⣿⣿⣷⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⠿⣿⡿⢿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡿⠟⣁⠀⠀⠀⠀⠀⠀⣀⠀⠘⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣏⠁⠚⠉⠀⠀⠀⠀⠀⠀⠉⠃⠀⠸⣿⣿⢿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢻⠋⠀⢠⣤⡀⠀⠀⠀⠀⠀⣾⣦⠀⠀⠙⠁⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⣸⠀⠀⠸⠿⠇⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⡊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⡀⠀⠀⠀⠔⠀⠀⠀⠀⠐⠀⠀⢀⢆⠠⡶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠐⡶⠖⠻⠁⢀⡀⠔⠈⠀⠀⡠⠂⠀⠈⢞⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⣠⠄⢴⡈⠠⠠⠠⣦⠒⠁⣀⠄⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠁⢿⣧⣾⡇⠀⠀⠀⢹⡖⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⡠⠄⠺⣇⠀⠀⠀⠀⠋⠈⠀⠀⠐⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢂⠃⠈⠁⣀⠠⠄⠛⠒⢺⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡏⠢⠐⠉⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣦⣤⣤⣤⣤⣤⣴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⢻⠛⠈⠿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠈⠀⠀⢰⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣷⣆⣀⡇⠀⠀⠀⡀⣠⣴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠶⢯⣷⣿⠟⠀⠀⠀⠀⢿⣿⣴⣛⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⡨⠊⠀⠀⠀⠀⠀⠈⢛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠂⠁
⣿⣿⣻⣻⣛⣛⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣟⣻⣿⣿⣻⣛⣿⣟⣟⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣠⢠⣄⡀⠀⠀⠐⠀⠀⠀⠀⠀⣠⢄⡀
⠀⢀⠀⠀⣄⠀⢠⢠⠀⠀⠀⠀⠀⠀⠀⠘⠹⠇⣟⡂⢰⣧⣆⣤⠀⣄⠀⠀⠐⠫⠏⣴⡶⢰⢷⣆⡀⠀⢀⡤⠒⢦⡀
⢀⣭⣤⠾⢳⣆⠉⠂⠀⠐⠀⡀⠀⠀⣀⣤⠀⠀⣴⡥⢞⢁⣯⠖⠀⡼⠋⣼⠙⠂⣨⠁⠃⠋⢉⡾⡽⢰⣦⠄⠀⠀⣷⢀⡀
⢟⡟⠉⠰⣾⡟⣢⣤⣶⣿⣆⢰⣿⠿⠛⠉⢀⣶⣷⡂⡀⢼⢅⣀⡀⠀⠈⡑⠂⠛⢻⠀⣼⣹⠎⠀⠀⠚⠀⠀⠀⠀⡿⢸⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⢈⣶⣤
⠀⡗⠀⠂⣠⣇⠹⢿⣷⣽⣿⣾⣿⣤⠶⠟⣿⣿⣌⢠⣿⠷⠾⠯⠝⢋⡃⡶⡄⣄⣀⡈⠁⠙⠀⠀⠀⠀⠀⠀⠀⢠⠁⡾⠀⣿⣷⢀⡀⠀⠀⠀⢈⣷⡾⣿⣿⢀
⠀⠁⠀⠀⢇⠀⠰⣿⣿⣿⣿⣿⡇⠀⠀⣾⣿⣟⡱⣟⣤⣀⠀⠀⠈⠋⠙⠙⠸⠏⣼⠃⡹⢣⢦⣠⣀⠀⡀⠀⡰⠋⢼⣗⣤⣿⡇⣸⣷⠀⠀⠀⠨⢿⡇⢸⣥⣾⣇
⣀⣀⣰⠶⣦⡀⠀⠻⠿⠛⠿⡉⠁⠀⠀⠻⣿⣿⣿⣾⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠈⠛⠷⢪⡷⠊⣡⠞⠛⣿⣿⡟⠰⡿⠃⠀⢰⡤⠀⢀⣉⣺⣯⡿⠁
⠉⠁⠀⠀⠈⠛⠶⠀⠆⢀⠀⢀⣠⣤⣴⣿⠾⠛⠏⠋⠉⠙⠂⠀⠀⠐⣤⣰⣆⠀⠠⢤⠄⢤⢪⣙⠛⠒⠶⠿⠿⢤⣴⣿⣯⣾⣟⠁⠀⣠⣿⡇⢸⣷⣬⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠈⠉⠀⠀⠀⠀⠀⠀⠠⢀⠲⣶⣶⣦⣴⣤⠀⢿⣿⣿⠧⣨⢒⣉⣤⣨⣴⡿⣀⣢⣴⣤⠄⠂⢠⣻⣭⣿⣾⣿⡛⢓⡾⠿⡿⣳⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠈⠓⠈⠀⠀⠉⠁⠙⠟⠿⠛⠊⠛⠖⠚⠀⠈⠁⠈⠙⠛⠛⠿⠛⠛⠮⠹⠃⠁⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠄⢀⢄⡸⠀⠐⢅⡤⡀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠑⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠄⠀⠀⠀⠀⠄⡻⠦⠙⠁⡠⠚⠒⡄⠈⠃⢖⢠⡀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠑⠀⠀⣔⠦⠊⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠑⠪⡢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡮⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢕⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠑⠉⡄⠀⢀⠀⠀⠂⠀⢢⠀⠀⠀⠐⠀⠀⠒⠀⠀⠀⠀⠆⠑⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡂⢠⡇⠀⠀⠀⠀⠀⠀⡠⠀⠀⠈⠙⠛⠶⣄⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠂⠀⠀⠀⠀⠠⠁⢈⠀⠀⠀⠀⠛⢈⠇⠀⠀⢄⠄⢀⠏⠀⠀⠀⠀⠄⠀⠀⠈⠃⠀⠀⠈⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⡀⡀⠁⠐⠚⠂⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⢀⡰⠂⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⡄⢴⠀⠀⠀⠀⠀⠀⡄⠀⠘⠀⠈⠁⠄⠠⠀⠠⠀⠠⠤⠡⠈⢢⠈⠢⠀⠀⠄⠁⠀⠀⠊⠀⠁⠌⠑⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⡄⠁⠀⠤⠀⠀⠢⡀⠀⠀⠀⠔⠁⡀⠀⠀⠀⠀⠀⢀⣤⣤⣀⠀⠀⠀⠀⠀⠈⠲⠤⡤⠤⠴⡊⠀⠐⠂⢬⢶⢰⠎⢠⠆
⠀⠀⠀⠀⠀⠀⠀⢠⡠⠁⠀⠀⠀⠀⠙⠒⡒⠂⢒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⠀⠀⠀⠀⠄⠀⠀⠤⠀⠈⠧⠀⠚⠏⢈⣟⠚⠛⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⢀⠀⠀⠀⠸⠁⠀⠁⡀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠐⠄⣠⠐⠓⡂⠀⠀⢐⠀⢀⠂⠀⠀⢄⠀⠙⠃⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠃⠀⠘⠲⠄⠀⠀⠀⢀⠀⠀⢀⡤⠊⠁⠀⢀⠂⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠀⠈⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠁⠀⠀⠀⠀⠨⠀⢀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠐⠐⠀⠃⠀⠀⠀⠀⠀⣆⠀⢀⡆⠀⠀⠀⠀⠀⠄⠀⠈⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠰⣀⠃⠀⠀⠀⠀⠀⡂⠀⠀⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠁⠈⠳⠀⠀⠀⠀⠀⠶⠂⠀⠰⠤⠀⠀⠀⠀⠠⠬⠉⠀⠉⢥⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢱⡸⠠⠐⢠⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠎⠄
⠀⠀⠀⠀⠀⠀⠀⢀⡔⢀⠀⢦⠉⠃⠀⡀⠀⠀⠋⢄⠀⢠⢄⠖⠀⠠⠤⠴⡠⢒⠢⡛⠫⠁
⠀⠀⠀⠀⠀⠀⠀⢿⠿⡂⠊⡁⠀⠀⣾⣿⡇⠀⠐⠅⢄⠀⡡⣶⢱⠀⠄⣤⣸⣾⣶⣉⠅⡦⡤⣀
⠀⠀⠀⠀⠀⡀⠀⡔⣑⠂⠹⢄⡀⠀⣈⠁⣀⡀⠐⡉⣬⡜⢶⡰⢏⠀⣻⣿⣿⣿⣿⣿⣮⠋⠌⣶⠄
⠀⠀⠀⠀⠈⣞⡄⣂⠀⠀⠀⡀⠂⣾⣿⣷⢟⣫⣖⠍⢪⠎⠀⣈⡒⠈⢻⡿⢿⣿⣿⡿⠦⣂⠀⠀⠀⣀
⠀⠀⠀⠀⢀⠻⣾⡿⠀⠰⢿⠀⢵⣿⣽⣾⣿⡿⢟⣽⢜⣀⠻⣫⣁⢤⡃⢀⣈⡥⠌⡁⢀⠀⢿⣿⣒⡾⡀
⠀⠀⠀⠀⠀⢞⡝⢳⠟⠜⠀⠀⣾⣿⠿⣛⣭⣶⣿⣿⡦⠃⣬⡠⡩⠃⠀⠀⢀⣀⡤⢌⠃⡁⠢⣙⠋⠈
⠀⠀⠀⠀⠈⠐⠐⢃⡉⣹⡤⢰⣿⣶⣿⣿⣿⡿⣟⣯⣾⣧⠋⡴⠈⠀⠉⠉⠉⠀⠀⢀⡇⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠜⠃⠀⠁⣾⣿⠿⣛⣫⣵⣾⣿⣿⡿⣫⣴⠥⠤⠤⠤⠖⠒⠊⠉⠀⢨⢀⠤⠀⣀
⠀⠀⠀⠀⠀⣀⢀⣀⠠⠴⠰⠷⠶⠛⠛⠛⠛⠋⠉⠁⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠐⠊⠉⠌⢀⠀⡛⠂⠂
⠀⠀⠀⠀⢨⣰⡅⠀⠀⠀⠀⠀⠀⡆⠀⡆⠀⢰⣦⠀⢰⡖⣤⢰⡖⣤⠀⢦⠀⠆⠀⠀⠀⠀⠀⠈⢰
⠀⠀⠀⠉⠃⠁⡇⠀⠀⠀⠀⠀⠀⡏⠉⡇⢀⠧⢿⡆⢸⡗⠉⢸⡗⠉⠀⠈⡏⠀⠀⠀⠀⠀⠀⠀⢸⠟⠂
⠣⣈⡣⠄⠐⠈⠁⠀⠀⠀⠀⠀⠀⠁⠀⠁⠈⠀⠀⠁⠈⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠐⠤⣠⠍⡀
⠟⠧⢀⡀⠀⠀⠀⠠⠤⣶⣶⣶⣶⣶⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠤⠤⢠⣦⣤⣤⣄⣀⡀⠀⠀⠀⠑⠥
⢌⠂⣡⠋⠀⠀⠀⠀⠀⠀⠈⣉⣛⠿⢁⠄⠠⡤⣀⢄⣀⠤⢀⡀⣠⢠⡀⠠⠄⠸⢛⣋⣉⠉⠀⠀⠀⠀⠀⢤⢜
⠿⠓⠁⢠⠤⡲⢶⣭⣍⣥⣧⣤⣥⣤⣤⢣⣭⠲⣭⠻⢠⣭⡐⣤⢮⣭⣤⡝⢦⣤⡨⢥⣤⣭⡰⢛⢇⠲⠦⠄⠃
⠀⠀⠀⠀⠘⠳⠄⣿⢿⣾⣏⣿⢾⣁⢿⣮⣿⣸⡏⢕⢱⠹⣷⢯⢸⣷⣾⠁⣾⣻⣧⢸⣿⣾⡇
⠀⠀⠀⠀⠰⠀⠠⠿⢘⠿⢿⢿⠾⢧⢺⠿⠻⢿⠙⠌⠈⠰⠿⡦⠸⠿⠿⠷⠯⠩⠿⠾⠿⠽⠗⡠⠄⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠉⠈⠁⠀⠀⠁⠁⠁⠀⠀⠈⠁⠁⠀⠈⠀⠀⠀⠈⠀⠉⠀⠈⠈⠈⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⣠⠖⠒⠒⢦⣀⣀⠀⠀⠀⢀⣠⣀
⠀⢰⠁⠀⠀⡞⠁⠀⠈⠱⡄⡴⠁⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⣇⠀⠀⠱⡀⠀⠀⢀⡿⡇⠀⠀⠀⢀⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠉⠲⡄
⠀⠀⠈⠲⣄⣠⠟⠢⣤⠞⢀⣧⣀⣀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠤⣀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠘⡆⢸⡄
⠀⠀⠀⠀⠀⠻⡇⠘⡟⠀⡽⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⡴⠉⠀⠀⠀⠀⠀⠉⠲⡄⠀⡇⠀⠀⠀⠀⠀⢠⠇⢰⡇
⠀⠀⠀⠀⠀⠀⠘⣼⣣⠞⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢜⣾⠁⠀⠀⣀⠔⢁⣠⠔⠀⣹⠀⠸⡄⠀⠀⠀⠀⠆⢀⡞
⠀⠀⠀⠀⠀⡔⠲⡿⠋⢠⠔⠋⠉⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⢴⣮⠭⣿⡖⣊⣉⡀⣈⡩⠤⠔⠊⠟⣆⠀⠘⢄⡀⠀⣀⠴⠋
⠀⠀⠀⠀⠀⢇⢀⣽⠀⡏⣠⠴⠖⠒⠒⠒⠒⠦⡀⢷⠀⠀⠀⢀⠀⠀⠀⠀⢯⡇⠠⠔⠲⠄⠀⠘⠉⠉⢸⠃⠀⠀⠀⢹⣻
⠀⠀⠀⣠⠔⣿⣿⠀⠀⣷⣟⠒⠒⠒⠒⠒⠒⠒⣇⣾⠀⠀⠀⠻⣉⡗⠀⠀⠀⠙⢄⠀⠀⠢⠤⠔⠂⣠⠏⠀⠀⠀⠀⢀⠧⣀
⠀⠀⠀⠁⠚⢹⢻⠀⠀⣿⢉⡭⡛⠒⠒⠚⠻⠥⠈⣽⡄⠀⠀⠀⠈⠳⢄⠀⠀⠀⠈⢑⡄⢤⠄⠐⠞⢹⣦⣀⣀⡠⠤⠺⣯⡼⠃
⠀⠀⠀⠀⠀⠛⠈⢧⠠⡻⡄⠀⢀⣀⠁⠀⣠⠀⣰⠃⠙⡄⠀⣠⡤⢤⠀⠙⠲⠤⣤⠚⠙⠪⡖⠐⡖⠁⠀⠙⣄⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠈⢳⣇⠈⠒⣤⣬⣭⠥⣴⣊⣁⣀⣤⡿⠔⠓⠤⠼⠃⠀⢀⠔⢧⠀⢀⠀⡇⠀⠱⣀⡤⠴⠉⣆⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢪⡀⠉⠉⡇⠈⠉⠀⠈⣧⣤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣄⡉⠉⢹⠧⠤⠔⢻⣀⡴⠚⠁⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢹⠇⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡲⠊⠀⡀⠀⠀⢻⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⣴⠛⣆⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣸⢧⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠘⠖⠋⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⡦⠇⠘⠦⡤⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠀⠀⢀⣀⣀⣀⣀⣤⣴⠾⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⣇⠀⠉⠉⠉⠑⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠯⣵⡇⠀⠀⠀⠾⣩⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀⠀⠀⠀⠀⠀⢸⡟⢹⣿⣿⣄⠀⠀⢀⣼⣿⣿⣷⡙⣧⣀⣀⣀⣀⠠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⡓⢤⣄⣀⣴⣿⡿⠾⠿⠿⠏⠙⠚⠁⠿⠿⣿⣿⣟⢻⣷⣭⣿⣯⣳⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⡿⣿⣦⣤⣼⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣿⣯⣝⠻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠹⣷⣄⣉⣽⣿⠇⠀⢀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣄⠀⠀⠸⡻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⠈⠛⠿⣿⠀⠀⠀⠀⠀⢔⣢⡀⠀⠀⠀⠀⣿⠀⠀⠠⠷⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡄⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⢤⣤⡤⠟⠁⠲⢶⣾⡿⠛⣧⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠈⣽⠋⠆⠀⠀⡀⠀⢀⠀⣸⡇⠀⢻⡀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⣾⠁⠀⠈⠉⠀⠈⠀⠀⢠⣿⡇⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⡰⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀⠀⢧⡀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⣰⠏⠀⠀⠠⢀⡀⠀⠀⠀⠀⠀⠈⢿⣷⡀⠘⣷⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣸⠃⠠⠃⠀⠀⠀⠀⠉⠁⠀⠈⠉⠀⠈⣿⣷⠀⠸⡆⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁⠀⠃⠀⠤⡀⠀⣀⡀⠀⠀⠀⠀⠀⣀⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣦⠀⢀⠔⠈⠀⠈⣿⣿⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⣿⣿⣿⠃⡄⠀⠀⠀⠀⠀⢹⣿⡆⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡏⠀⠖⠀⠀⠀⠀⠀⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡟⠀⠀⢸⣀⠆⠀⠀⠀⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢸⡿⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠁⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⡒⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⠁⠂⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢠⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⢸⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢄⠀⡴⢦⣠⠞⢆⣠⠶⡄⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⡏⠲⡎⠀⠙⢧⡰⠃⠀⠈⠳⣠⠞⠁⠈⡦⠚⢳⢀⣀
⠀⠀⠀⠀⠀⠀⠀⣤⢤⣼⠈⠉⠓⠼⣀⡤⠴⠖⠒⠚⠛⠛⠒⠒⠶⠤⣄⡹⠔⠚⠉⢸⣀⣠⡄
⠀⠀⠀⠀⠀⣤⠄⠸⣀⡀⠀⡠⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⡀⠀⡁⣀⡇⢤⡄
⠀⠀⠀⢠⡀⢹⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⠀⠀⢰⡁⣀⡀
⠀⠀⠀⠈⣆⣀⠆⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⢄⢀⡜
⠀⠀⠫⠀⠀⠀⡔⠀⢀⣠⠤⠤⠶⠦⠤⢤⣀⡀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⠶⠤⠤⢤⣀⠀⠘⡄⠀⠀⠨⠃
⠀⢶⠊⠑⡀⡘⠀⠚⠉⢀⣀⣤⠤⢤⣀⡀⠀⠉⢤⠀⠀⠀⢀⠌⠁⠀⣀⣠⠤⠤⣄⣀⠈⠙⠂⠘⡀⡐⠁⢒⡆
⢀⡤⠓⠈⢠⠃⠀⠀⣚⡭⢤⢶⡖⢶⢦⢭⣁⠀⠈⠆⠀⠀⠆⠀⢀⣩⢥⢶⠖⣶⢦⠬⣝⡂⠀⠀⢡⠈⠐⠪⣄
⠈⢲⣄⠀⢸⠀⠀⠈⠩⣒⠬⢧⣭⣥⡮⠤⠚⠑⠀⠸⠀⠸⠀⠐⠛⠢⢬⣦⣭⣥⠯⢔⡪⠉⠀⠀⢸⠀⢀⣴⠋
⠺⣅⣈⡆⢀⠀⠀⠀⠀⠈⠙⠒⠒⠒⠒⠋⠁⠀⠀⢨⠀⢰⠀⠀⠀⠉⠓⠒⠒⠒⠚⠉⠀⠀⠀⠀⢀⠐⢧⣀⡽
⢠⠞⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⠀⠈⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠙⢦
⠈⢙⠶⠤⡘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠑⠒⠛⠦⠞⠓⠒⠁⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⡠⠴⣎⠁
⠀⠓⠢⡼⠁⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣃⣤⣖⡺⢦⠾⣒⣦⣀⣻⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠹⣤⠜⠃
⠀⠀⢾⣀⣀⣀⠱⡀⠀⠀⠀⠀⠀⠠⠞⠁⢻⡉⠀⣉⣓⣋⡁⠈⣹⠃⠙⠂⠀⠀⠀⠀⠀⠀⡰⢃⣀⣀⣨⠄
⠀⠀⠀⢰⣃⣈⡇⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⢉⠀⢃
⠀⠀⠀⠀⠀⠸⠀⢀⣀⡑⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⡁⠀⠀⢸⡍⠈
⠀⠀⠀⠀⠀⠉⠉⡏⣀⣼⠀⠙⠳⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠚⠁⢸⡀⢸⡉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠁⢹⣀⢤⠖⠻⡈⠉⠛⠒⡶⠶⠦⠶⠶⡒⠚⠋⠉⡸⠓⣦⢄⣸⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠸⠶⠚⢧⢀⣴⡊⢙⣆⠀⢀⡞⠉⡷⣄⢠⠟⠦⡇⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠳⠊⠈⠳⠏⠈⠾⠁⠈⠉

⠀⠀⠀⠀⠀⠄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⡰⠢⣀⠜⢣⣀⡔⢦⠀⣠⡄⠀⠀⠀⠄⠀⠀⠠⠄⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢰⠛⢢⡎⠀⠙⢧⣠⠏⠀⠀⠹⣄⡼⠋⠀⢹⡴⠚⡇⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⡏⠉⠛⠦⠼⢁⣠⡤⠴⠷⠒⠒⠒⠒⠾⠦⢤⣄⡈⠧⠴⠛⠉⢹⠀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣇⠀⡇⠀⣠⠴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠦⣄⠀⢸⠉⢠⣇⣀⡀
⠀⠀⠀⠀⠀⣀⣘⡇⠀⠉⢉⡵⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢮⡉⠉⠀⢰⠃
⠀⠀⠀⠀⠀⢯⠉⢱⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⡎⠉⣹⠃
⠀⠀⠀⢲⡚⠋⠉⢉⡞⠁⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠈⢳⡉⠉⠙⢓⡖
⠀⠀⣠⠤⠷⡄⢀⡞⠠⠴⠚⠉⠁⢀⣀⠀⠀⠉⠙⠲⣄⠀⠀⠀⠀⣀⠖⠋⠉⠀⢀⣀⠀⠈⠉⠓⠦⠄⢳⡀⢠⠾⢤⣀
⠀⠀⠈⣣⡤⠟⣼⠁⠀⠀⠴⢚⣉⣉⣉⣉⣙⠲⠄⠀⠘⣆⠀⠀⣰⠃⠀⠠⢖⣋⣉⣉⣉⣉⡓⠦⠀⠀⠀⣷⠻⠤⣴⠋
⠀⠐⢏⠁⠀⢠⡇⠀⠀⠖⢯⣉⢹⡘⠦⢏⡼⢉⣳⢦⡀⢸⡀⢀⡇⠀⡴⣞⡉⢯⡹⠴⢃⡏⣉⡽⠲⠀⠀⢸⡀⠀⢀⡽⠃
⠀⣠⡔⠛⢦⣸⡇⠀⠀⠀⠙⠲⢭⣙⣛⣛⣩⠭⠄⠀⠀⠈⡇⢸⠁⠀⠀⠰⠭⣍⣛⣛⣋⡭⠔⠋⠀⠀⠀⢸⣇⡴⠛⠦⣄
⠀⠈⢙⡶⠋⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠙⢶⡊⠁
⠀⠰⢯⣀⡀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢾⣑⣣⡀⢀⣜⣊⡷⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⢀⣠⠽⠂
⠀⠀⣰⡋⠉⡷⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢁⣀⡍⢩⣄⡈⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢾⠉⠱⣄
⠀⠀⠀⢉⡟⠁⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣦⡾⠭⢭⣛⣋⡭⠭⢷⣴⢧⡀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠈⢻⡉⠁
⠀⠀⠀⠚⠒⣦⠤⢤⠳⡄⠀⠀⠀⠀⠀⠈⠁⠀⠸⢤⣠⠤⠶⠶⠤⣄⣤⠇⠀⠈⠁⠀⠀⠀⠀⠀⢠⠞⡤⠤⡴⠒⠓
⠀⠀⠀⠀⠰⠧⠤⡾⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢧⣀⣽
⠀⠀⠀⠀⠀⠀⢰⣃⣀⡤⠤⡝⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢫⠤⣄⣀⣘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣁⡤⡇⠀⠈⠙⠲⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠶⠋⠁⠀⢸⣄⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠤⢶⠋⢹⡄⠈⠉⢙⣟⠒⠒⠒⠒⣻⡋⠉⠁⢠⡏⠙⡶⢤⣼⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠖⠋⢧⣀⠴⡏⢈⣧⡀⠀⣼⠁⣹⠦⣀⡜⠑⠾⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⡀⠀⠀⠀⠀⢄⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⣀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⣀⢀⠀⠠⠤⠤⠀⠀⠀⠀⠀⠔⠆
⢀⡀⠀⠀⢠⠆⠀⠀⠀⢀⠀⠤⠀⠖⠂⠀⠀⠈⠉⠁⠒⠒⠊⠋⠁⠀⠀⠁⢸⠠⠤⠐⠒⠈⠁⠀⠁⠈
⠆⠁⠔⠀⠘⠀⢀⠠⠀⠤⠤⠤⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡄⠀⠴
⠀⠐⠈⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢀⠀⡀⠀⠈⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠂⡀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠠⠀⡔⡈⠰⠄⠀⠁⠌⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠤⠤⠤⠐⠒⢒⡔⠃⠀⠈⠐⢲⡊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢀⠇⢸⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢈⠀⠸⠀⠀⠀⠀⠄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⢈⡀⠀⡆⠀⢀⣀⣀⣀⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡆⠉⡋⠉⠀⠀⠈⠩⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠇⠀⠀⠀⠀⠀⣃⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠙⠹⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⡄⠄⠀⠀⠀⠀⠀⠀⢀⠒⠈⠀⢘⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⣀⣀⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠿⠛⠋⠉⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠪⣶⣄⠀⠠⠀⢀⡞⠁⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠰⣿⣿⣴⣣⡗⠋⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⢀⢔⡿⣿⣿⡿⠧⢄⣀⣀⡀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⣒⡵⠛⠁⠀⠈⢣⡀⠀⠀⠉⢉⡽⠃
⠀⠀⠀⠀⠂⠉⠁⠀⢱⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠏⠀⠀⠀⠀⠀⠈⠗⢄⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⢿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⡀⣂⣀⣀⠀⢠⣷⡀⠀⠀⣀⣤⣤⣴⡾⠿⠛⠉⠀⢸⡇
⠀⠀⠀⠀⠀⠈⠙⣿⣷⣿⣿⣷⡿⠿⠛⠋⠁⠀⡀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⠀⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠸⢻⣿⢿⡇⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⠛⠁⠀⠀⠀⠀⠀⠑⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠈⢢⣀⠀⠀⠀⠂⠀⠀⠀⢀⣤⣿⣤⣄⣀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠓⠒⠊⠀⠀⠀⢠⣾⠟⢻⠁⠈⠙⣏⠢⠀⠀⠀⠀⠀⠀⠑⢦⡀⠀⢀⡠⠞
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢻⡇⠀⠀⠀⠀⠀⢸⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⢈⡇⠀⠀⠀⠀⣰⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠐⠂⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠲⣶⠆⠀⠀⠀⠀⠀⠀⠀⠤⣴⣿⣧⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠉⠙⢧⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⢀⣴⣿⡟⠃⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠛⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⢰⣀⡆⣰⣆⢰⣲⣴⣖⡆⢆⡖⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠆⠀⠀⠀⠀⠘⠀⠃⠋⠙⠘⠉⠘⠋⠀⠘⠀⠀⠀⠀⠀⠰⠆⠀⣀
⠀⠀⠀⠀⠀⠰⠆⠀⠁⠀⠀⠀⠀⠀⠀⠀⣾⣆⢸⡇⣾⣛⡃⢻⡀⣿⡄⣾⠀⠀⠀⠀⠀⡀⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⢦⢠⡀⣸⠀⠀⠀⠀⠀⣿⠈⢿⡇⣿⣉⡁⠘⣿⠃⢷⡏⠀⠀⠀⠀⠀⣇⢠⠄⡴
⠀⠘⠀⠠⣤⣀⡘⠮⠃⠃⢁⣀⠀⠀⠀⢠⡀⢀⡄⣠⣤⡄⠀⣠⡀⠀⣤⣤⡀⠀⠀⠀⣤⣀⠈⠈⠔⢁⣠⠤⠆⠀⠁
⠀⠀⠀⠀⠀⠀⠉⢀⣤⣾⠿⣛⡀⠀⠀⠀⢷⡾⠀⣿⠤⠄⢰⣏⣧⠀⣧⣤⠇⠀⠀⢠⣝⠻⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠻⢋⣵⠞⣫⡤⠀⠀⠀⠸⠃⠀⠻⠶⠆⠟⠉⠹⠂⠇⠘⠇⠀⠀⠶⣍⡛⢦⡙⠋
⠀⠀⠀⠀⠀⢠⡄⠀⠀⠑⠚⣥⠖⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡓⢮⠙⠂⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣠⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠀⠒⠒⠊⠁⠀⠀⠀⠀⠉⠑⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠄⠀⠀⠀⠀⠀⠀⠈⠀⠐⠂⠄⡀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⡠⠂⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠖⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠐⠀⠠⠁⠀⠀⠀⢠⠦⡀⠀⠐⠠⠀⡀⠀⠈⠉⠁⠀⠀⡩⠀⠀⠀⠀⠀⠰⠢⠔⠘⠐⢃⡮⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠈⠑⠂⢄⣄⣀⣀⢀⣀⣠⡔⢺⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⣠⣶⠤⢤⢀⠀⣀⣠⡛⢡⡌⢊⡷⠊⠀⣋⠏⠀⠀⠀⠀⠀⢀⠎⠀⠈⠁⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢒⢢⠤⡴⣽⠀⠶⠀⡩⠗⢀⠉⠶⡤⠤⣢⠃⠀⠀⡇⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠆⠀⠈⢓⡒⠚⠋⠀⠠⠴⠀⠀⠀⠮⠹⢀⡜⠀⠀⣀⠤⠒⠁⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣢⡀⠧⠔⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠪⠗⠲⡍⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⣠⡜⠢⠤⣄⣀⣀⣀⡀⠤⠴⠊⢡⠃⠀⢀⡁⠘⡀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢰⠸⢀⡀⠀⠀⠑⢄⠈⢢⠄⠲⠀⠃⡤⠔⠉⢡⠀⢃⢀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠊⠆⠀⠈⡗⠂⠐⠁⠁⠺⡀⠀⡏⢠⠂⠀⠀⠀⡀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠸⠀⠀⠀⠀⠀⠀⠙⠢⡝⠞⡐⠁⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠄⠀⡆⠀⠀⠀⠀⠀⠈⠄⡇⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⡰⡇⠀⠁⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⡌⠀⡇⠘⠀⠀⠀⠀⠀⠀⠀⠸⢀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠰⠇⢀⠀⠇⠀⠀⠀⠀⠀⠀⠀⢠⣮⢀⠀⠠⠤⠤⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠆⠰⡘⠾⠁⠀⠀⠀⠉⠀⣀⣀⡀⠷⠠⠤⠤⠀⠒⠠⠜⠒⠀⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠄⠀⣸⠀⠀⠈⢻⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⢢⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠐⠞⣰⢠⣄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⢈⠂⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠢⢄⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢱⡋⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠑⠶⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠒⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⢀⣠⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣪⡖⠁⠙⢦⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡜⠃⠀⣰⡀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⡴⠏⠱⡄⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠂⠀⠀⣨⣵⢦⣀⢠⣿⠂⠸⡇⣸⡄⢸⠀⠸⣷⡄⢠⢶⡍⡔⠀⠀⠰⡦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⣁⣴⢦⡀⠀⠹⠦⣄⢹⡎⠀⣴⠼⡇⠹⠉⠺⠵⣃⠈⢷⡇⡠⠾⠃⠀⢀⠴⣄⣈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⣈⣁⡤⢾⡆⠒⠚⠻⢿⡧⢜⡙⠢⣰⡶⢶⡁⠘⢛⡤⢼⠾⠛⠒⠒⢰⡧⡄⣨⢁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣭⣥⢿⠀⢻⡄⠀⣄⠑⢮⣓⢀⡿⢳⡞⢇⢀⣫⡴⠃⣰⠀⣴⠏⠀⣷⣬⣭⣯⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡒⠦⠝⠿⢦⣼⡟⠏⢸⠀⠈⠀⢀⠉⣻⣴⣿⣷⣼⡟⠉⡐⠈⠆⠰⠀⡕⢻⣦⠴⡟⠫⠶⡲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣰⠀⢤⣦⣄⡈⠻⢷⣼⠆⠀⣴⠖⠹⠅⣿⡿⢻⡟⣿⣟⠸⠝⠛⠆⠀⢱⣧⠿⠅⢁⣤⣴⡄⠈⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⠐⠳⣁⠔⢐⠏⢳⠀⠠⢻⠦⠔⠺⢂⣀⡼⢿⣟⣈⣠⣻⢿⢆⣀⠲⠛⠤⢴⡏⠀⢠⡏⢸⡈⠀⣪⠟⠂⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰⠂⠀⡿⠛⠣⡀⣸⠂⣾⠉⠀⣤⣄⠀⠋⠀⠠⠈⢿⡟⠁⠔⠀⠃⠀⣧⡄⠀⠉⣆⠂⡅⢠⠟⠛⣏⠀⢲⡆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣦⡈⠛⢿⣧⡘⢖⢡⣛⣸⠉⠀⠠⣿⣿⣦⡴⣿⠷⣦⣜⣠⡴⠿⡷⢤⣴⣻⣿⠄⠀⢙⣠⣳⡈⡾⢁⣞⠷⠋⢀⣼⡀
⠀⠀⠀⠀⠀⠀⢀⣬⡀⠙⢧⣦⡀⠈⠻⣾⡏⣼⡟⡃⠀⠀⢻⣿⣿⠐⡯⠀⠀⠻⠏⠀⠀⣿⢀⣿⣾⡟⠀⠀⢹⣻⡆⢿⣶⠟⠁⣀⣴⡿⠋⣨⣅
⠀⠀⠀⠀⠀⠀⣨⣀⠉⢲⠜⠋⠉⠢⡀⢻⠇⠸⣧⠙⡆⠀⠀⠉⠹⢆⣗⠂⢰⡏⢻⡄⢀⣿⡸⠋⠉⠀⠀⢺⢃⡾⠃⢸⡏⣠⠞⠉⠙⢦⠚⠁⣀⣇
⠀⠀⠀⠀⠀⢀⣀⡈⢙⡎⠰⣣⠀⠀⠙⣾⠇⠀⠙⣷⡟⠓⢦⣴⡤⠞⢻⣦⠌⢳⡟⢀⣾⠟⠣⢤⣤⡴⠚⣿⡟⠁⠀⢸⣷⠁⠀⢀⡛⡀⣷⠋⣁⣀⡁
⠀⠀⠀⠀⠀⢁⣀⣉⣙⠁⢸⢹⣗⡠⡀⢹⡅⠀⠀⠈⣧⠀⠂⢷⣄⣠⣾⡟⢷⡈⢀⡾⣻⣷⣀⣴⠟⠐⢠⡟⠀⠀⠀⣹⠃⠠⣐⣽⡏⡇⢈⣋⣁⣀⡈
⠀⠀⠀⠀⠀⠀⠬⣥⣤⠤⠸⡘⣯⢻⠿⣻⣧⣦⡄⠐⠘⣦⡀⠈⠙⣿⡃⠀⠊⢻⠟⠀⠀⣛⡟⠏⠀⣀⡾⠁⠀⠀⢀⣽⣿⡿⡻⣿⣰⢁⣿⣉⣉⡉⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡴⠦⡘⣝⢮⠛⣿⣿⠀⣿⠀⠀⠀⠹⢦⣔⣧⣿⡈⠂⡠⡀⠲⣀⣧⣿⣠⠶⠋⠀⠀⠀⢠⠞⠐⠛⣟⡵⡳⢡⠞⠦⣄⠀⢙
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡄⠸⡷
⠀⠀⠀⠀⠀⠀⠀⣿⣿⢰⣶⢀⠰⣶⡄⣿⣿⣏⣴⣶⠀⣤⠖⢲⣶⡆⢰⣶⡠⢶⡶⢺⣿⡗⢲⣶⡄⠐⣶⣦⠀⣠⡶⠒⣶⣦⢰⣶⡀⠉⠀⢠⣴
⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⠃⠀⣿⡇⣿⣿⢷⣌⠉⢸⣿⣀⣸⣿⡇⢸⣿⠁⠀⠀⢸⣿⡇⢸⣿⡟⠀⣿⣿⢸⣿⣀⣠⣿⣿⢸⣿⡇⢀⠃⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠈⠛⠀⠀⠙⠃⠻⣿⠀⠙⢷⡜⠻⠟⠁⠙⠃⠈⠛⠀⠀⠀⢸⣿⡇⠈⠛⠃⠀⠘⠛⠈⠻⠟⠁⠘⠛⠈⠻⠿⠋⠀⠈⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢠⠀⣠⡀⢀⣀⡀⣄⠀⣀⣀⣀⣀⣄⣄⠀⡀⢀⠀⣤⡄⠈⠙⠃⣀⣄⣄⣀⣀⣄⣀⠀⡄⢀⢠⡀⢠⡄⢀⡄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢎⠳⣙⣿⢶⠞⢡⠇⠀⣿⠿⠛⡷⠟⠻⣷⢆⢠⠀⠘⠃⠀⠄⣱⣻⠞⡳⣿⠛⠟⡧⠀⢳⡘⢶⢷⡿⡉⢊⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠶⠷⠞⠁⠀⠀⠈⠓⠚⠡⣸⡇⠈⣿⣽⡐⢠⠠⣣⣿⡾⠀⢸⢁⠜⠓⠚⠁⠀⠀⠙⠲⠷⠶⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⢋⡼⠿⣴⡷⢿⣦⠙⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⠅⢾⠙⠀⢹⡇⠀⢟⡆⢩⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠄⢾⡅⠀⠳⠞⠀⢬⡧⠨⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⣲⡈⢳⡥⠸⢂⢬⡟⢡⡚⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣭⡄⠛⣾⣿⠋⣠⣽⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠂⠈⠉⠉⠉⠀⠈⠴⠆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠠⠀⣤⣀⠐⠆⢄⠂⠀⠀⠀⢀⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠈⠉⠁⠀⠑⡐⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠊⠀⠀⠀⠀⠀⠀⠂⠀⠈⢦⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⢰⣧⠀⠀⠀⢰⣦⡄⠀⠀⠀⠑⡦⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡗⠀⠀⠀⡀⠀⠀⢀⠀⢨⠀⠁⠀⠀⠀⠀⠀⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⣦⡀⠀⠀⠉⠁⢠⡄⠀⠀⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠙⢥⠀⠺⣶⡶⠟⠁⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠄⣀⠈⠉⠁⠀⠀⢀⡠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⢉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠘⠡⠀⠤⠃⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⣀⠤⡀⠈⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠒⠒⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠐⠄⠀⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠞⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⡀
⠀⠀⠀⢤⡀⠀⢀⠠⠘⠈⠀⠀⠐⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⢄⠀⠀⠀⠃
⠀⠂⠐⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⠀⠂
⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠠⠤⠄⠤⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡁⠀⠀⠀⠀⠀⣀⣠⡀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠲⡄⠀⠀⢨⠁⠀⡅⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⠰⠶⠇⠀⠀⢸⠀⠀⡇⠀⠀⠀⡇⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠀⠀⠇⠀⠀⠀⡇⠘⠒⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢤⢀⣠⡇⠀⠀⡄⠀⠀⠀⠀⠀⠀⡇⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠑⠒⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⡅⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠃⢲⠂⠋⠀⠀⠈⠳⠒⠲⠶⠶⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠐⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢸⡀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠘⠈⠁⡇⠀⠀⠀⠀⠉⠉⢋⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠎⠀⠀⢀⠀⠀⠀⣰⠀⠀⠀⡠⠀⠀⠀⠀⠈⢑⡦⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣤⣀⣀⣀⡀⢀⡀⣀⣠⠇⠀⢰⡋⠀⠀⠀⠀⠉⠀⣀⡠⠃
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣴⣾⣿⣯⣶⣶⣶⣶⣷⣷⣷⣶⣿⣿⣶⣧⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⠁⠉⢉⡉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⢀⠄⢸⡀⡀⢀⢀⠄⡀⠀⢀⡏⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢀⠀⣀⢄⢠⢰⣉⠀⡇⢁⢘⠀⣙⢈⣢⢸⠁⢸⢸⢸⠒⠁⠀⢸⠃⠀⡇⡈⢁⢀⠄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⠀⢸⠀⠘⠌⠠⠀⠁⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠊⠈⠤⠌⠘⢶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⡤⠤⡤⠜⠙⠉⠓⢲⣀⠀⠀⠀⠀⠰⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢀⣀⡀⢀⠔⠃⠀⠀⠀⠀⠀⠀⡤⣀⡀⠀⠀⠋⠙⠊⠳⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣃⠔⠋⠁⠀⠀⠀⠀⢨⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠠⡈⢩⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠁⠀⠀⠀⠀⠀⠀⠀⢐⠉⠀⢀⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠠⢌⣒⡲⠄⢺⠁⠀⠙⠀⠀⠀⣠⣀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠢⢤⡀
⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⣄⣀⠦⠬⠭⠹⣇⠀⠀⠀⢀⡠⠟⠁⠘⠤⡄⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠅
⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⣸⢸⠀⡝⣿⢿⠀⠀⠀⠸⢤⣀⡶⠞⠀⠀⠀⠀⠀⠈⠉⠓⠋⠒⠩⡃⠀⠀⠀⢦⣀⠀⠀⢀⡹
⠀⠀⠀⠀⠀⠀⠈⡄⠀⢀⡏⡼⢠⠃⠉⠀⠀⠀⠀⠀⢸⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢅⠀⠀⠀⠀⠀⠀⠀⢨⡇
⠀⠀⠀⠀⠀⠀⠀⢣⠀⠈⢰⠃⡾⠀⠀⠀⠀⠀⠀⠤⡄⡄⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⡆⠀⠀⠀⠀⠀⠀⢤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⢀⠜⠀⠀⢴⠆⠀⠀⠀⠀⢁⠡⡀⠀⠀⠀⠠⣞⡀⠀⠀⠀⠀⣤⠀⠀⠈⢆⣀⠀⡀⢀⡖⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⡀⠀⠀⠀⠀⠀⠀⠀⠉⢁⣠⢷⢤⡀⠀⠀⠠⣀⠄⠀⠀⠀⠀⠀⠀⢀⠴⠩⢋⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢷⣦⠤⠤⠤⢤⢴⣖⠉⠁⠴⡄⠸⣗⡲⠤⠄⢤⡤⠤⠤⠤⢶⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠡⠀⢻⡇⠀⠀⡾⠞⠉⠒⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠧⠴⠟⠁⠀⠀⠟⠃⠰⢷⠄⢀⡀⠀⢤⠄⢢⢄⡘⠚⠷⡀⠀⠀⠐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠶⠦⠀⣀⡀⠈⠙⠀⢸⠀⠀⠉⠉⠂⠼⠟⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡏⠐⣠⠄⠀⠉⠀⡒⠦⢼⣀⢀⠀⡀⢀⠀⡀⠠⣐⠢⠱⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠖⠩⠧⠤⠾⠦⡤⠴⠾⠳⠈⠀⠑⠚⠂⠈⠁⠈⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⡀⢠⠀⠀⠀⢨⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⢀⠀⠈⡀⠀⠀⠈⢂⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⡄⠌⠸⡄⡤⠂⠀⠀⠐⠤⢠⠃⠈⠢⡄⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠠⢄⠀⢠⣀⠧⡀⠀⠀⢠⠲⠴⠀⠀⡠⠆⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⣶⣦⣾⣿⡿⠋⢻⣝⣿⣿⡿⢳⣿⣿⣿⣿⣿⡿⠟⠉⠀⠈⠀⠀⠘⠂⠙⠋⣠⣿⣿⣶⣶⡴⠠⠸⠃⠹⠠⠌⢀⠀⠀⠀⠀⠀⠀⠁
⣀⡁⠙⠛⡗⣼⠷⣿⡏⠀⢀⢸⣿⣿⣟⠛⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠛⠿⠘⠇⠀⢸⠀⡀⠈⠀⠀⠀⠔⢤⡈⣢⠀⡄
⠀⠀⠀⠁⡆⠙⠛⠛⢱⠄⠀⢀⠙⢿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠙⠉⠉⠀⠀⠀⠘⠂⠀⠐⠀⠀⠀⠀⠀⢀⠘⢋⠁⠀⠉⠙⢳
⣤⣀⠤⠄⡧⠀⠀⠀⢀⠀⠤⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠂⠀⠀⠎⢠⠄
⠿⠧⠤⡃⠂⠈⠀⠐⠂⠀⢀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢘⠠⡭⠄
⠀⠚⠒⠀⠃⠀⠀⢠⠴⠶⢈⠩⠀⠀⡈⠀⠀⢠⠔⠐⣀⣠⣘⢁⡄⠀⠀⠀⠓⠉⠀⠀⠀⠀⠀⠀⠐⠂⠘⠁⠀⠀⣾⠀⠉
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠠⠀⠀⠀⠀⠀⠳⣤⣿⠋⠉⢹⡄⠂⠀⢀⡠⠤⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡸⡆⢀⡀
⠀⠀⠀⠀⠀⢀⠀⠐⢄⠀⠀⠀⠠⠄⠁⠀⠀⠚⠃⢈⣥⣤⠖⡀⠀⡐⠉⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠂⠁⡇
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢂⣀⠀⠀⠉⠘⠂⠈⠠⣠⠛⠁⠀⠡⠀⠁⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⡠⠞⠉⠀⠀⠀⠀⠡⠁
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢳⠄⡀⣠⠀⠀⠈⢍⠁⠠⠀⢢⠄⢀⡈⢄⠀⠈⠀⠀⠀⣠⠊⠀⠀⠀⠀⠈⠂⡀⢰⡭⠄⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠈⠃⠀⢩⢆⠠⠀⠀⣄⠐⠂⠀⠀⠤⠉⠀⡀⠀⣼⠁⠀⠀⠀⠀⢠⡞⠒⢷⡨⣯⣒⣒⠢⠌⠁
⠄⢀⡄⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⣀⠀⠀⠀⠀⡂⢨⣡⠴⠥⢢⢞⣡⣚⣛⡃⡛⡀⠀⠹⣄⠀⢠⣤⣴⣿⣷⡄⠠⠈⠒⠠⠤⠐
⠓⠋⠉⡒⠦⣄⠀⠀⠀⠀⠉⠁⠀⠀⠈⠓⢄⡀⡖⠂⠘⠁⣀⣠⣦⣤⣾⣿⣿⣿⠁⢁⡀⠀⠙⠶⣿⣿⣿⡿⣝⣀⠈⠀⠀⠀⠀⠓
⠀⠔⠈⠀⠆⡈⠳⡀⠊⠀⣀⣤⡴⠤⢀⡀⠀⠙⠢⣾⣾⣿⣿⣿⣿⡿⠿⡿⡟⠹⠐⠈⠟⠀⠀⠀⠀⠈⠀⠀⠚⠠
⠀⠀⠀⠄⠈⠠⠀⢸⣦⣶⡟⠁⠀⠀⠀⠑⡄⠀⠱⡙⡛⠻⠟⠙⠠⡆⡸⠚⠀⠀⠀⠀⠀⢀⣀⣈⠑⢒⣠⠶⠋⠀⣄
⠀⠀⠀⠀⠀⠀⠐⠀⡇⠈⠁⠀⠀⠀⠀⠀⢳⠀⠀⠃⠈⠀⠀⠀⠀⠃⠀⡀⠃⠀⠀⠀⠀⠤⣄⣠⣶⣿⣿⣦⠲⠄⠠⠄
⠀⠐⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⡀⠀⣠⡿⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣀⡤⠀⠁⠐⡎⠛⠋
⠁⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠊⢘⣵⣾⡿⡇⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠂⣡⣤⣤⣄⣀⡤⠀⠀⠠⠇⠘⠿⠃
⠀⠐⠀⠀⠀⠀⠀⠀⢀⠀⢢⡀⢿⠛⠻⠕⠀⡴⢀⣁⢠⡴⢾⣶⣶⠒⣉⣁⣀⠀⣀⣙⣛⡻⣿⡿⠃⠀⠴⣰⠀⢆⡦⠠⣴⡆⠀⢆
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠂⡀⠱⣞⣀⣀⣴⢺⣷⡌⣿⡿⠃⣸⣿⣿⠀⣿⡿⣿⡇⣿⡟⣛⢣⣯⣀⠀⠀⠀⣠⢤⠀⡄⣤⡀
⢀⠞⠀⠀⠀⠀⠀⠀⠀⢈⠄⠂⡌⠀⢸⣿⣈⣉⡁⣿⣷⡄⣿⣿⣿⡄⣿⣷⣿⠇⣿⡿⣿⣧⠉⠉⠱⡀⠀⠁⠉⠈⠈⠉⠀⠀⡀
⠃⠀⠀⠀⠀⠠⣀⠀⠀⠄⠚⡁⠀⠀⢸⣿⢻⣿⠇⢿⢏⠁⣿⣧⠿⠇⢿⡏⢿⠇⣤⡄⣿⣿⠀⠀⠀⠤⠀⠒⠤⡤⠤⡀⠀⠀⠈
⣤⣤⣤⣔⣶⡯⠬⣵⠁⠉⠁⠀⠀⠀⠸⣿⣾⣿⡆⣿⣵⣶⣿⡏⣵⣧⣿⡇⣿⡅⢽⣧⣿⡟⠀⠀⠀⠀⠀⠀⠀⠘⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣼⢇⢻⠀⢹⡞⢸⡼⢪⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠀⠀⠃⠈⠈⠁⠀⠀⠈⠁⠋⠀⠀⠀⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠠⠄⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⣀⡀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠁
⠀⠀⠀⠀⠀⠀⣠⡴⠚⠉⠁⡹⢀⣤⡀⠀⠉⠀⠀⠠⢀⣀⠀⠀⢠⡀⠀⠘⠁⠀⢀⠇
⠀⠀⠀⠀⣠⡾⠋⠀⠀⠐⠋⣰⠋⠀⠀⣀⠀⠀⠀⠰⠂⡀⠉⢱⠟⠲⠦⠴⠶⠚⠁⠸⠀⠀⠀⡀
⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⣰⠃⢠⡆⠀⢑⡦⢀⡼⠀⢸⡋⢠⠋⣴⠀⣴⢂⡄⢀⡴⢚⡁⠀⢸⡃
⠀⠀⠀⠸⣇⠀⠀⡀⠀⠀⠏⠀⠘⠀⠀⠘⠀⠘⠁⠐⠾⠃⠇⠘⠁⠈⠁⠺⠁⠘⠁⡘⠃⠐⠼⠃
⠀⠀⠀⠀⠈⠉⠁⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⡀⠀⠀⠀⠀⠈⠙⠋⠛⠛⠀⠀⠀⠀⠀⠀⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢥⡶⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⡷⠌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠸⠚⡖⠀⣶⢰⠀⣾⡀⠈⠎⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⠀⠀⠀⠀⠀⠠⠄⠀⠀⢀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠐⢐⠂⠣⠊⡐⡂⠀⠀⠀⠀⠀⠀⠀⠣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠉⡐⠰⢀⠆⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠭⠂⠀⡀⠥⠡⣠⠅⠐⠄⠀⠡⢊⠀⠀⠀⠀⠀⠀⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠣⠍⠈⠤⠄⠌⡀⠄⡍⡀⠀⠐⠭
⠀⠀⢀⡀⠀⠀⠀⠀⠀⠠⡋⠔⠐⠈⣊⡸⠀⠀⢈⠀⠀⠀⠀⠀⠀⠃⠀⠀⠄⠀⠀⢄⡠⠀⠈⠀⠀⢈⠄⠢⢀⠔⠄⡒⡁⠀⠀⠀⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠠⠂⡍⡘⡀⠀⢤⢣⠁⠀⠀⠀⠀⠐⠊⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠁⠖⡀⠊⠉⠀⢐⣀⡄⠀⠀⠀⣀
⠀⠀⠀⠀⠀⢈⠀⡁⠀⠀⠀⠐⠂⠑⠂⠘⠃⠀⠀⠀⠤⡀⢀⠡⠤⠤⠤⠤⠄⠄⡀⢀⠤⠀⠀⠀⠁⠐⠂⠠⠌⠢⠀⠀⠀⠀⢈⠀⡁
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⡰⠈⠁⠠⠀⠀⠀⠀⠀⠄⠀⠑⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠃
⠀⡀⠀⠀⠀⠀⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠈⠂⠊⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠀⠀⠀⠠⢄
⠀⠐⠂⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡐⠀⠀⠀⡀⢰⠀⣀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠂⢄⣀⡠⠐⡆⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⣙⣞⢾⣉⠀⠀⠀⢰⠂⢄⣀⡠⠐
⠀⢰⠀⠀⠁⠚⠀⠓⠀⠁⠐⠀⠀⠀⠀⠀⠀⢤⠀⣠⠀⠀⠀⠀⠠⠄⠤⠀⡄⠀⠈⠀⠀⠀⠀⠀⠀⠁⠸⠀⠉⠀⠀⠂⠈⠀⠚⠀⠓⠈⠁⠀⠆
⠀⠈⠂⠀⠀⠙⠀⠃⠀⠀⠘⠀⠀⠀⠀⠀⠓⢺⣻⠥⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠⢂⢐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠙⠀⠃⠀⠀⠐⠁
⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠘⠃⠸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡇⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠆⠤⠀⠀⡄⡀⠀⠀⠀⠀⠀⠀⡠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⡔⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠄⠄⠓⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠆
⠀⠐⠐⠁⠀⠀⠀⠀⠀⠀⠐⢈⠀⠀⡁⠀⢰⠀⡆⠀⠀⠀⠠⠂⠐⡀⠄⠀⠀⢤⠀⠀⠠⡐⠀⠄⢀⠄⠀⠀⠀⠈⠍⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠘⠀⠈⠁⠀⢸⠀⠁⠀⠀⣠⠁⠼⠆⢱⠀⠨⠐⢀⠀⠭⠀⡀⠈⢀⠂⠀⠀⠀⠀⠈⠁⠐⠂⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡀⠀⠀⠀⠀⠀⢸⠀⡇⢁⠀⣁⡠⠒⠂⡀⡄⠰⠂⢃⢀⠂⠈⢸⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠂⠐
⠀⠀⠀⢀⡠⠠⠀⠣⡈⠀⠉⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠉⠁⠀⠀⠀⠀⠀⢈⠁⠁⠀⠀⠀⠀⠀⠀⠂⠈⠀⢉⠀⠀⠀⢄
⠀⠀⠀⠄⠀⠀⠤⢂⡊⠀⡒⢄⡀⠀⠆⡄⠀⠀⢸⠀⠀⢀⡃⢸⢀⣀⣀⡍⠈⢀⠃⠘⠀⠁⠈⠀⠀⠀⢀⠀⠃⢀⡠⠐⠃⠘⡐⠠⠆⠀⠠
⠀⠀⠈⠐⠈⡆⡀⠈⢁⡘⠋⠀⢠⠀⠘⡄⠀⠀⠨⠀⠠⡀⠀⢸⠀⡤⢀⠀⠀⠛⢠⡄⠃⠀⠄⠀⠀⠀⢠⠃⡠⢄⠀⠘⠃⡈⠂⣀⠐⠁⠀⠂
⠀⠀⢀⠑⠤⠆⠁⣁⠁⢃⠘⠛⡐⢀⠈⠀⠀⠀⠐⠀⠂⠈⠂⠸⠀⠀⠀⠌⠢⠤⠌⠠⠀⠌⠀⠀⠃⠀⠀⡉⡀⢢⠘⠃⠀⠈⣀⠉⠐⠤⠂⡀
⠀⠀⠈⢀⠀⡞⠢⣉⠀⠑⡃⠤⠈⠀⠔⠀⠀⠀⢠⠄⢂⢀⠤⠖⠒⠒⢒⠀⡠⠤⡀⠐⠀⠐⠠⡀⠀⠀⠀⠱⠀⠁⡤⢀⢂⠀⣉⠠⠒⠀⡀⠁
⠀⠀⠀⠀⠒⠁⠔⠂⠀⠃⢄⠰⠠⠃⠀⠀⠀⠀⠀⠂⠀⠃⡰⠃⠘⠒⢋⠠⠀⠄⠐⡆⠀⠅⠀⡁⠀⠀⠀⠈⠘⢄⠅⠀⠜⠀⠐⠂⡀⠒⠁
⠀⠠⠂⠢⠀⠀⠈⠑⠄⠁⠈⠀⠀⠀⠀⢠⠄⠀⠀⠀⡇⠀⠀⡀⠘⠂⠒⡇⡈⠉⠀⢁⠠⡄⠘⠄⠀⠀⣠⠀⠀⠀⠀⠀⠈⠣⠈⠉⠀⠀⡄⠉⠂
⠀⠈⠂⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠈⠁⠉⠉⠃⠀⠀⠑⠊⠁⠀⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠲⠂⠈⠉⠉⠉⠉⠉⠉⠒⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⢂
⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡁⠀⠁⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠸⠇⢀⠀⢀⡠⠄⢲⡾⠉⠉⠉⠉⠉⠉⠉⠀⠒⠒⠂⠠⠤⢈⣳⠀⠀⠀⠀⢄⠀⠀⠃
⠀⠀⠀⠀⠆⠀⣾⠞⠁⠀⠀⢈⡴⢂⣈⠁⠢⡀⠀⠀⠀⠀⡠⢊⣡⡌⠑⢤⡈⠑⠤⣀⠀⠀⠀⢸
⠀⢀⡤⠞⠂⢴⡇⠀⠀⠀⠰⠋⣰⣿⣯⣴⣦⣸⡆⠀⠀⢠⣧⠼⠿⠷⠷⠤⣵⠀⠀⠘⡷⠀⠀⢸
⠀⢸⡇⠀⠀⢾⠀⠀⠀⠀⠙⠋⠉⠁⠀⠀⠀⠈⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸
⠀⠀⠙⡗⠒⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣀⠸
⠀⠀⠀⢇⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⣠⡔⠀⠀⠀⠀⠀⢀⡏⠀⠉⢿
⠀⠀⠀⠘⢧⡀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠘⣏⠃⠠⠴⠦⠖⠘⢋⡼⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠈⠳⢤⣀⠙⢦⡀⠀⠀⠀⠀⠀⠈⠑⠦⣤⣤⣤⠴⠋⠁⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⣐⠂⠲⡄⠀⢀⠔⠁⣠⠞⣠⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠳⢦⣀⡀⠀⠀⠀⠀⠀⠴⠄⠀⠀⠀⢀⣠⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⢈⣤⡁⠀⠘⠣⠚⠁⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠶⡶⠶⠶⣶⠶⠶⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠓⠀⠀⠰⣉⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣸⠇⠀⠀⢷⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡷⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠋⠁⠈⠯⣀⠀⠀⠀⢀⣨⠋⠈⠁⠀⠙⠛⠒⢶⡆⠠⠤⠴⠶⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⢤⣤⡤⣤⠞⠓⠒⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣯⠤⠤⣠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⣀⣸⡧⣤⣤⠤⠤⠴⠶⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⢀⠘⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡘⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⡐⠁⠀⠀⠘⢄⢸⣧⣤⣤⣤⣤⣤⡤⠶⠶⠶⠶⠶⠶⠶⠶⢶⡿
⠀⠀⠀⠀⣌⠁⠀⠀⠀⠀⠈⠻⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢀⠀⢃⠀⠀⠀⠀⠀⢀⡴⣦⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⢤⣀⡘⠀⣴⠀⠘⣧⣈⡄⠀⠀⠀⢀⡏⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠉⠛⠦⠻⡤⠚⠀⠈⠀⠀⠀⠀⣘⠁⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠶⠒⠀⠀⠀⡟⠀⠀⠀⢸⡄⠀⠀⠠⠴⠶⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⣀⣀⢸⣇⣀⣀⣀⣀⣀⣀⣀⣰⠇⠀⠀⠀⠀⠀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣇
⠀⢀⠴⠚⠓⠠⣄⠀⠀⠀⠀⠀⠀⠀⡌⠉⢻⡄⠀⠀⠀⠀⠀⢸⠇⠀⣠⣤⠤⠴⠦⠄⢈⣦
⡰⠃⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠙⢒⠦⣷⠀⠀⠀⠀⠀⢸⡖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⣧⣀⣀⣀⣀⣀⣀⣀⣘⣁⢀⠀⠀⠀⢘⠋⠀⣼⠀⠀⠀⠀⠀⠸⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⣤⣤⣤⣤⣬⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣽⣿⣷⣶⣶⣶⣶⣶⣾⣯⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠛⠙⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣧⡁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢡⣽⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⡏⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡙⡙⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⢏⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠒⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣿⣿⣿⣿⣿⠃⣿⣿⣿⣿⢿⣿⣿⣿⠿⢿⣿⣿⡿⢿⣿⢿⡿⢿⡟⠻⠉⠉⠃⠀⠀⠀⠀⠺⣄
⠀⠀⠀⠀⠀⠀⢀⡞⠉⠀⠀⠿⡿⢿⣿⢃⡎⠁⠈⠀⠁⠀⠈⠀⠁⠀⠀⠉⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢣
⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠉⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹
⠀⠀⠀⠀⠀⡝⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡅
⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⡠⠤⠔⠊
⠀⠀⠀⠀⠀⠈⠹⣄⣀⣆⣀⡀⠀⢠⠴⠁⠸⠤⣆⣠⣀⡰⠤⠶⣤⠴⠦⠞⠢⠤⠴⠒⠛⠒⠚⠒⠚⠶⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠛⠁


⣿⢿⠿⠿⡿⡻⢿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⢻⣿⣿⡿

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢶⣾⣿⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣟⣯⣽⣿⣦⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢨⣽⡿⣳⣶⡾⣿⣶⣿⣾⣭⣭⣻⣿⣷⢾⣷⣷⣾⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣷⣦
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠿⢿⣿⣿⣿⠿⣷⣶⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠐⣨⣿⣟⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⡄⢸⣿⣿⡿⣶⣤⣙⠻⣿⣦⣀⠀⠀⠀⠀⠀⢀⣤⣤⣶⣶⣿⣿⣧⣄⡀
⠀⠀⠀⣠⣴⣤⣽⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣧⣿⣿⣿⣿⣾⣝⡻⣷⣜⠻⣿⣶⡀⠀⠀⠀⠈⣹⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⡞⢰⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣼⣿⣿⣿⣼⣿⣿⣿⣿⣿⣞⣿⣧⠹⣿⣷⣄⠀⠴⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢳⣜⢿⣿⣿⣿⣿⣧⣤⣄⣉⡛⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣦⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠐⣼⣿⣷⡾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡹⢋⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢀⣿⣿⡾⣖⠀⣾⣿⣿⢹⣿⣷⠻⣴⣿⣿⣿⣿⣿⣿⣿⠋⡄⢿⣿⣿⣾⣿⣿⣿⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠠⢻⣿⣿⣣⠈⣟⣿⣿⡜⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠛⣿⣯⣿⣆⠋⣻⣿⣿⣌⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢹⣿⣷⣿⣦⡔⠻⢿⣟⣷⣾⣿⣿⣿⣋⣻⣿⣿⣿⢿⣿⡿⣹⣿⣿⣿⣿⣿⣿⡿⣿⢁⣿⣿⣿⡿⢿⢿⣿⣿⣿⣿⣿⣿⠛⠻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠉⠛⣷⣿⢟⣾⣀⣻⠟⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⠟⢿⠟⠻⣿⣿⣿⣿⣿⣿⣶⣶⣿⠛⠾⣭⡻⣿⡿⠀⠀⠈⠙⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠑⢿⡟⣽⡟⣾⣿⣾⡏⢻⣿⣽⣿⣿⣿⣿⣿⣽⣿⣾⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣤⠈⠙⣮⢧⡀⠀⠀⠀⠀⠈⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠣⠿⢿⣿⣧⣌⣛⣹⣿⠾⣿⣯⣟⢻⣿⣙⣾⣿⡿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡘⠹⠏⢿⣷⣼⢯⣧⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⣶⣾⣛⣛⣛⣛⣫⣗⣿⣿⣿⠄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⣀⣽⢻⣷⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⣿⣿⣻⣿⣿⣿⣿⠸⣿⠉⢻⣿⣇⠀⠘⢿⣿⣿⣿⣿⣏⣉⣉⡿⢼⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣾⢿⣿⣿⣿⣿⡄⢿⡄⠘⣿⣷⡀⠀⠈⣿⠙⣿⣿⣿⡿⣧⡀⠀⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢼⣯⣿⣿⣟⣷⡈⠻⣦⣸⢿⣳⡀⢤⣿⠇⡽⢿⣿⣿⣟⣿⡀⢉⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠾⠿⣦⣈⠛⠿⠿⣿⣶⣩⡰⠃⠀⠙⠛⠿⢿⡇⠈⡾⣷⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠶⠟⠑⠿⢳⣶⣤⡇⣠⢆⣠⠀⡀⣸⣿⣿⡿⠩⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⠃⠻⠟⡿⠹⡿⢿⡿⣍⡁⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡟⠇⠀⢿⣿⣦⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣴⡴⡄⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣽⣷⣿⣿⣶⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣥⣀⡩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙

⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⢀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⠄⢀⡀⠀⠀⠀⠀⠀⢀⠀⠤⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡤⢤⠄⡀
⠀⠀⠀⢀⠜⠁⠀⢀⣀⠀⠀⠙⢆⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠑⡀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⢸⣀⡀⠐⠁⠀⢑⠀⠀⢸⠀⠀⠀⠁⠀⠀⠀⠀⠸⡄⠂⠌⠀⠀⠀⠃⠀⢀⡐⠁⠈⠃⠀⠀⠀⠀⠀⠐⠴⠐⠐⠁⠀⠑⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡈⠀⠀⢸⠒⠄⠠⠀⠀⠀⠁⠄⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠌⠀⠀⠀⢸⠀⢀⠀⠀⠀⠀⢚⠀⠀⡂⠀⠀⠀⠀⠀⢀⠔⠁⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠱⠤⣀⠀⠀⠁
⠀⠀⠀⠀⠀⡀⠈⠀⠀⡠⠂⣀⣀⠀⠀⠘⡄⠂⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⠀⠀⢀⡄⠁⠀⠀⠀⠀⠀⣀⠤⠄⠀⠀⢀⡀
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢣⠔⠈⠀⠁⠉⠅⠀⣰⠃⠀⠀⡎⠀⠀⠀⠀⠈⠀⠀⠀⢥⠀⠀⠸⡄⠀⠀⠀⠂⠉⠀⢀⡄
⠀⠀⠀⠘⠥⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠮⠤⠤⠤⠐⠁⠀⠀⠀⠣⠤⠤⠤⠤⠄⠀⠀⠤⠊⠀⠀⠀⠈⠒⠂⠤⠄
⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠠
⠀⠀⠀⠀⠠⠠⠁⡌⠀⢀⠀⠀⢠⢤⠀⠀⡔⠀⠀⢀⡆⠰⠀⠀⢐⠐⠒⠠⠀⢀⠒⢒⠒⠢⡀⠐⣲⠀⢠⠀⠂⠀⠀⠠⠡⡄⠄⡠⠀⠄
⠀⠀⠀⢠⢀⡑⠆⠃⠰⠀⣠⠄⠀⢐⠀⠀⡃⠐⠀⢐⠰⠆⠤⠀⠃⠰⣈⠂⠘⢨⠀⠩⠝⠀⠅⠆⠸⣀⡌⢠⠃⠐⠀⠀⢓⠠⡀⢘
⠀⠀⠀⠐⡀⡅⠠⠨⢈⡁⠀⠀⠀⠈⠂⠖⠀⠀⠀⠀⠈⠁⠈⠀⠇⢠⠄⠀⠂⠨⠀⡤⠀⠈⠀⠈⠢⢤⠀⡀⠀⠀⠨⣀⣁⠆⠐⠈⡀⢀⠄
⠀⠀⠀⠈⠀⢠⠂⠘⠀⠠⡘⠀⠅⠘⠁⠑⠀⠀⢐⢀⠎⢂⠀⡆⠃⣠⠀⠀⠀⢸⣀⠄⠀⠀⠀⢂⡐⠊⡠⠃⠀⠀⠀⠀⠀⠀⠣⠀⠠⠂⠤
⠀⠀⠀⠀⢀⠐⠀⠀⠑⠀⠀⠀⠐⠂⠀⠀⠒⠁⠈⠉⠀⢀⠉⡐⢈⡁⢀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠪⠈⢠⠤⠂⠒⡀⠁⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢈⡀⠆⠠⠁⠁⠀⠀⢀⣀⠀⠀⠀⢀⡀⠀⠀⠀⢀⠌⣀⠴⠊⠀⠀⢀⠠⠀⠆⢉⠂
⠀⠀⠀⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⢀⠄⡫⠀⠂⠐⠄⠀⡌⠀⠃⠀⠀⡂⠀⠀⠀⡰⠃⠘⠁⠀⠀⠀⠀⠀⠈⠀⠂⠂⠁
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠈⠀⠀⡁⡈⡄⠀⠀⠀⢠⠀⠠
⠀⠀⠀⠀⣤⠈⡀⠀⠀⠀⢀⠄⠠⡀⡠⡀⠀⠀⣀⠀⠀⠨⠄⠀⠀⠀⠀⠀⠈⢆⡀⠈⣁⠀⠀⠀⡠⠤⠤⡀⠀⡀⠤⠤⠀⠀⡤⡄⡀⡀
⠀⠀⠀⠀⠀⠀⢠⠂⠀⠐⢡⡈⡀⠀⠁⠀⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⡘⠀⠯⠂⡐⠐⠀⠀⠀⠀⠀⠀⠂⡠⣈⠕
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠈⠄⠙⠀⠂⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡀⠀⠄⠁⢀⡔⠊⠀⡇⢨⠀⠀⠁⠀⠀⠒⠁
⠀⠀⠀⢅⠀⠀⠤⡠⠀⠐⠀⠀⠩⠂⠀⠀⣄⠀⠀⢀⣠⠂⠀⠀⠀⠀⠀⠀⠀⠤⣀⣉⡠⠎⠀⠱⣄⣉⡤⠃⠱⣀⡠⠠⡱⠀⡄

⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠁⠀⠠⠁⠀⠀⠀⠀⠐⠊⠀⠀⠀⠀⣄⠀⡴⡀⠸⠟⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈⠠⡀
⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠐⡀⠄⠠⠀⠀⠋⠀⠐⠀⠀⠄⠀⠠⠀⠀⠀⠀⠀⠀⠀⡀⠸⠈
⠀⠀⠠⠀⠀⢈⠒⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⣀⠀⠀⠀⠀⠀⠈⡀⠀⠀⢃
⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠉⠀⠀⠄⠀⡀⠀⠄⠀⠈⠀⠀⠀⠀⠄⠀⠀⠀⡀⠐⡀⠀⠈⡄
⠀⠀⢃⠀⠀⠀⠀⠀⢱⡀⠈⠀⠀⡤⡄⡀⢄⢀⠀⠀⠀⣠⠀⢎⠉⢂⠀⠀⠀⠈⢠⠂⠐⠀⢀⡈⠀⢀
⠀⠀⠈⠢⡀⡀⠀⠀⠀⠣⠀⠜⢰⠀⠀⠑⠂⠙⠈⠒⠈⡘⡀⢠⡍⢘⡆⠀⠀⢀⢀⠀⠀⠁⠉⠁⠀⠐
⠀⠀⠀⠀⠈⠐⠂⠀⠐⠁⡌⠀⠈⠧⠜⡁⡟⣷⠀⣊⡩⠻⠕⠛⠓⠹⣀⠆⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠤⠃⠀⠐⡬⡂⠀⠀⠄⠀⠂⢀⠬⣤⢀⡄⠀⡀⢠⡖⢰⡀⠀⡀⠀⠐⠀⠀⠀⠀⠄⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⢂⢸⠠⣩⠸⠁⠀⠃⠮⠄⢸⡀⠎⢁⡀⠀⠀⠠⠀⠐⡐
⠀⠀⠀⡠⠦⠀⠄⠂⠐⠀⠀⠔⡁⠘⣶⠀⠀⠀⠈⠁⠀⠀⠽⠑⢀⠀⠀⢀⠀⠉⠀⠀⠀⠰⠂⠁⡀
⠀⠀⢀⡄⢀⠀⡀⡀⠠⢤⠀⠀⠠⠰⠀⠀⠀⠨⣤⠔⠀⡀⠀⠐⠂⠁⠀⡚⠐⠀⠀⠀⠤⠄⠀⠈⢀⠰
⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠉⠀⠀⡆⠀⠀⠀⠀⠀⠉⠒⠒⠀⠀⠀⠐⠂⠚⠃⠘
⠀⠠⠗⠄⠢⠢⠐⠌⠊⠒⠉⠀⡀⠀⠀⠈⠀⢀⡀⠀⢀⠅⠀⠀⠠⠄⠀⠈⠐⠀⠀⠀⠀⠀⠐⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠠⠄⠸⠧⠶⠏⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⡤⠶⠶⠒⠒⠒⠂⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⣠⣄⡀⠓⡾⠁⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⣻⠃⠈⢹⡞⠁⠀⠀⡠⠁
⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⢀⠎⠀⠉⢲⡿⠟⠁⠀⢠⠎
⠀⠀⠀⠙⠶⣄⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⣀⡠⠤⠦⠤⠤⣄⠀⠀⠀⠀⠀⢈⣁⣀⣤⣉⣀⡀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠟⠛⠋⠉⠉⠙⠻⢿⢦⡀⠀⠀⠀⠀⠠⠖⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⣴⠖⠀⠀⠀⠀⢀⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠤⠂⠺⢶⣶⣶⣦⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⣤⠀⢶⡸⡏⢹⡇⠀⠴⠶⠄⣀⠀⣤⠄⣀⣀⠄⣀⡀⠀⡀⠀⠀⢀⠉⠀⠀⠈⠉⠉⣽⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⢹⡇⠘⡧⠁⠘⡇⢠⠄⠀⠀⣿⠀⡏⠀⣀⠉⢢⡀⠈⠻⣶⣇⡰⠋⢶⡄⠀⠀⠀⢰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠘⣷⠀⠱⠟⠀⡇⠈⠠⠟⠀⣿⠀⡷⠀⢉⣀⡼⢁⣀⣠⠟⠛⢁⣴⠟⠁⠀⠀⢀⣿⠇⠀⠀⠰⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⢿⡀⢰⣶⠀⠃⡇⢀⡄⠀⣿⣤⠁⢸⡿⡇⠀⢹⡏⢹⠀⣠⡿⡏⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠘⡇⠈⢻⡄⢰⣇⣸⣧⣀⢸⡏⠀⢸⡇⡇⢠⣞⠀⢸⢠⡟⢀⠃⠀⠀⠀⢀⣿⠁⠀⠐⠒⠒⠒
⠀⠀⠀⠀⠀⠰⠆⠚⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⢻⡀⣾⠡⣼⣍⠁⡬⣥⡄⠁⠶⠿⠀⠷⣾⡿⠀⣌⣸⡇⢸⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠈⠁⠉⠀⡇⠘⢦⡇⢻⢈⡏⢩⣵⣆⠐⣦⣤⡄⢨⣭⡀⣆⠀⠀⠀⢠⡛⠃⠀⢻⡆
⠀⠀⠀⢀⡠⠄⠂⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⡷⢸⡌⡇⠘⣧⢸⢸⠃⣼⡷⢃⡿⠁⣉⣤⣐⠒⠚⠇⠀⠀⣼⠇⠀⡀
⠀⠔⠊⠀⠀⠠⡶⢦⢀⣴⡾⣶⣄⠀⠀⠙⣿⣦⠀⠀⠀⠀⠀⠀⢁⢰⣦⠀⢸⠃⡇⠸⣇⡰⠈⡀⠛⢁⡾⠁⠀⣏⡀⠀⣙⣷⠀⠀⡼⢠⣷⡾⠁
⠀⠀⠀⠀⠀⠘⠧⠞⢤⣿⠀⠈⣷⡇⠀⠀⣷⠃⠀⠀⠀⡤⠤⠤⡜⢰⡏⣣⣼⣇⡃⠀⣿⡇⣸⡇⢀⣾⢡⡤⠤⠬⠭⣙⠉⠁⠀⠀⠁⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠶⣋⠀⠀⠀⣿⠀⠀⠀⠀⢳⡀⠀⢳⠾⠷⡇⠀⣿⠻⠶⠟⠿⠇⢧⡾⠃⢸⡇⠀⣀⠀⢹⡇⠀⠠⠄⠸⣏⠀⠀⢐⢛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠁⠀⠀⠀⠘⣷⡀⠀⠀⠀⢷⠀⠀⢳⣤⠀⢰⣿⠖⢒⣀⠀⡔⠒⠲⡆⢸⠁⠈⠙⣀⣼⠇⠀⠀⠀⠀⢻⡄⠀⠈⠉
⠀⠀⠀⠀⠀⢠⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠘⢣⠀⠀⠛⠀⣾⢿⠀⢿⣅⢰⠁⡀⢸⡁⡆⠀⠀⢸⡏⠁⠀⠀⠀⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⣿⠀⡀⣤⠟⣾⢰⡇⠈⡅⡇⠀⡄⠘⣿⠀⠀⠀⠀⠀⠀⢠⡄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠸⡇⢻⠀⡇⢘⡁⠀⢃⡇⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠄⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⡇⠘⢻⡅⣾⣿⠀⢸⠃⠀⢃⠀⣿⠀⠀⠀⠀⠀⢠⠟⠉⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⠄⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠁⠸⠛⠿⡥⠿⠘⠷⠖⠛⠶⠟⠐⠿⠀⠀⠀⢀⣰⠫⠊⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⢻⣆⡠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠿⠛⠉⠀⠀⣠⡮⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠙⠷⣎⣠⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠁⠀⠀⠀⢠⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⢀⣉⣀⣰⣶⣤⣴⠾⢿⣤⣀⢀⣀⣠⣶⠀⠀⠀⣀⡴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠈⠉⠉⠛⠛⠉
⠀⠀⠀⠀⠀⠺⠃⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠠⣄⣀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠋⠁⠈⢹⡣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠘⢇⣴⠖⠶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠊⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠸⢯⠖⠀⠀⠀⠀⠀⠀⠀⠐⣿⠗

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢤⣀⠀⠀⠀⠀⠀⠀⣠⠴⢚⡉⠉⢉⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣤⡀⢀⣞⣡⠖⠋⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⣦⣤⣰⡏⢯⠛⢿⡶⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠺⠽⢿⠀⠈⢧⠀⠙⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⣇⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⢸⡄⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢀⣀⣀⣠⣧⣀⣀⣀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠉⠀⣀⡀⡇⣀⠀⠈⣙⣦⣸⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠉⠉⢁⠔⠋⠉⠀⠀⣇⠀⠀⠈⠉⠓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣰⣧⠀⠀⠀⠀⠀⢀⡏⠀⣠⡄⠘⠒⠛⠒⠃⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⠉⠉⢉⠏⠉⠉⠓⢦⡀⠀⢸⡁⢸⠁⡇⠀⠀⠀⠀⠀⠀⠀⢸⠆⢸
⠀⠀⠀⠀⠀⣸⠁⢀⣀⠀⠲⠤⠿⠒⠀⠀⠀⠀⢹⣀⣈⣇⠈⠧⠇⠀⠀⠀⠀⠀⠀⠀⠈⢀⡏
⠀⠀⠀⠀⠀⡏⠀⡞⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⢻⠀⠳⣼⠀⠀⠀⠀⠀⠀⠀⢠⠀⣸⠃⠀⠀⠀⠙⠲⢤⣀⣀⣀⣠⠤⠚⠁
⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⣠⠎⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀⣀⣀⣀⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣒⣉⣉⣭⢵⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢴⣄⣀⣀⣀⣀⣀⣀⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⠭⣦⣤⠤⠒⢂⣈⠴⢦⡖⢤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⣁⡤⠤⠶⠦⠶⠤⠤⣀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠱⡀⠣⣀⠿⠗
⠀⠀⠀⠀⠀⠀⠀⢀⣯⠖⣩⠴⠒⠉⠉⣀⣀⣉⣉⠒⢬⣑⣺⠀⠀⠀⠀⠀⠀⢀⡀⠆⠃⠈⠀⡇⠀⠀⠀⠀⠀⢀⡀⠠⢄⣀⡀⠀⠀⠈⠙⠙⡆
⠀⠀⠀⠀⠀⠀⠀⢸⣡⠊⢁⡤⣖⣋⡩⠤⠴⠦⣀⠽⠲⡞⣇⠀⠀⠀⠀⠀⡀⠂⠁⠀⠀⠀⡠⠁⠀⣠⠴⠒⠉⠀⠀⠀⡀⠀⠈⠙⢦⠀⠀⠀⣰⢤
⠀⠀⠀⠀⠀⠀⠀⠈⣇⡾⠋⣠⠤⠄⠀⡀⠀⠚⠉⠂⠀⣷⠏⠀⣀⣀⡀⠀⠀⠀⠐⠂⣴⠞⡧⠀⢞⠁⠀⠀⠀⠀⢀⡦⠵⢄⡒⠰⣬⡇⠀⠀⠷⠚
⠀⠀⠀⠀⠀⠀⠀⠀⢏⢳⡀⠁⠀⠀⢀⠀⢠⠄⠀⠀⣸⠇⠀⣸⣡⡤⠟⠀⠀⠀⠀⠀⠈⠱⡇⠀⣾⠒⡲⠖⠈⠉⠀⠀⠀⠁⠈⠁⢸⣹⢤⣀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⢄⡀⠀⠀⠉⠁⠀⣠⠴⠃⣠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣤⢿⡀⠀⠀⠀⠀⢠⢤⡀⠀⠀⠀⢸⠁⠀⣸⡵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡝⠙⢒⠒⡟⣯⠶⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⠉⠢⡑⢤⡀⠀⠀⠈⠲⠁⠀⣀⡴⢷⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠉⠒⠼⣾⠤⠼⣦⠠⢾⣉⣉⡹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠚⠛⠯⣉⣱⠒⠿⠋⢯⡥⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣊⠀⠀⠀⠀⠀⢘⣦⡤⠼⠓⠶⠤⠤⢤⡄⠀⠀⠀⣠⣞⣉⡽⠇⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠘⡆
⠀⠀⢰⠒⢦⡀⠀⠀⠀⠈⠉⡳⠒⠂⢹⠧⣄⡀⠀⠀⠀⠀⠀⠀⡟⠒⠈⠉⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡁⠀⠀⠀⠀⡇
⠀⠀⠀⠑⠦⣼⠂⠀⠀⠀⡴⠁⠀⢠⡏⠀⠀⠈⠉⠙⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡟⠻⡖⠛⠁
⠀⠀⠀⠀⠀⠈⠳⡀⠠⡞⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠙⠦⠜⢦⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢰⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣄⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣿⣷⣿⣿⣤⣄⡀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⣀⣴⣶⣾⣿⣿⣷⣶⣤⣄⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⢀⡿⠋⠉⠀⠉⠀⠀⠀⠀⠃⠀⢰⣿⡟⠉⠉⠛⠻⠛⠛⣿⠀⠀⠀⠀⠀⢻⡿⠋⠉⠉⠀⠉⠙⢻⣿⣿⠃
⠀⠀⠀⣀⠀⠈⡝⠀⠀⠀⠀⣀⣀⠀⠀⢀⠀⣸⣿⡆⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠸⣿⣏
⠀⠀⠘⠋⠉⠠⣿⣄⠀⠀⠀⠉⠉⠀⠀⡈⠀⣏⡃⠀⠀⠐⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠉⢠⣠⡀⠁⠀⠀⠸⠂
⠀⠀⠀⠀⠀⠀⠪⠙⠻⠿⣶⠀⡶⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠐⠂⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠤⠁⠀⠡⠘⠁⠀⠀⠀⠀⠀⠀⠀⡄⢀⠂⠊⠀⠐⠀⠀⠀⠀⠈⠀⠀⣠⠃⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⣀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡤⡄⠀⠀⠀⠀⠀⠄⠀⡈⠀⠀⠀⠀⢢⢀
⠀⠀⠀⠀⢠⢨⡇⠀⠀⠀⣀⠀⠈⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠙⢇⠀⠀⠀⡀⠀⠀⠔⠳⠀⠀⠀⠀⠊⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠔⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡂⡂⢐⣐⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠑⠀⠀⠀⠀⠠⠔⠃⠃⠀⠀⠀⠀⠀⠈⠙⠒⠂⠈⠛⠍

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣭⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣯⣿⣿⣽⣿⣯

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣤⣤⡄⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠻⠦⠉⣤⣞⣹⣷⣄⣻⣷⣶⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⠾⢿⣷⣿⠇⢻⢈⣿⣿⣿⣿⣁⢠⣴⣆⣴⡗⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⢛⢿⣿⣋⢋⡜⣏⡻⠻⡯⢽⣶⣶⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠻⣶⣿⣿⡇⠀⢀⣠⠤⣾⠒⠋⠁⠀⠀⠀⣰⠆⠀⠀⣸⠁⡾⠢⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠢⣿⣿⣿⣷⡀⠈⢀⣀⠀⠀⠀⠀⠀⠀⣨⠁⠀⠀⠀⠹⣌⢁⣄⢟⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⣧⣾⣏⣰⠓⣽⣿⣿⠆⠶⢀⡄⠀⠀⣹⠡⣄⣠⣀⠀⠉⠘⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠛⠷⡠⣶⣿⣼⣿⣷⣔⣿⣷⣶⠰⠏⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠩⣿⣿⡿⢻⡍⣽⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⢀⣀⡜⣿⡗⠞⡉⡿⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣰⣺⣩⣿⣿⣽⣿⣿⡿⢷⣤⣿⣿⡟⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡵⢻⣷⣿⡿⢿⡍⣽⣿⣏⢨⠗⢾⣿⠿⠿⢿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⣵⣿⣿⣿⠒⠊⠉⠉⠙⠋⠀⠐⠉⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠠⡔⢲⡿⢻⣿⠉⣿⢸⣿
⠀⠀⠠⣤⣤⣾⣧⠚⠋⠀⢈⣿⣉⣡⣼⣋⣤⠀⠀⠀⢀⣀⣀⠀⠀⠀⣠⣶⣷⠄
⠀⠀⠀⠈⠈⠻⠿⢄⠀⠀⠀⠚⣿⠾⢿⣿⣿⡉⢋⣼⡟⣹⣿⣉⢋⡬⡟⠙⢻⣦⣢⣀⣠⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠀⠀⠨⠣⢿⣿⣿⣿⣶⣿⣟⢛⣿⣿⣶⣿⣿⣧⠀⠙⠋⢽⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠀⠛⠺⡿⣿⠛⢹⣄⣿⠀⠀⣈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣴⣿⡉⣃⡤⠃⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⣐⣤⣷⣾⠿⣿⢹⣻⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠟⢹⡿⡿⡏⢽⣿⣷⠰⠋⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣸⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠾⠉⣡⣼⣋⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠚⠿⠛⠛

⣿⠿⠿⠿⠿⠻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠁⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠻⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠛⠉⠉⠁⠀⠉⠉⠉⠙⠓⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⢤⠀⠀⠀⠀⠀⢀⣴⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⢀⡴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢻⡄⠀⠀⠀⠀⠀⠠⠶⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢴⣶⣾⣿⡿⠶⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⢻⡛⠶⣤⣀⡀⠀⠀⠀⢀⣀⣤⣾⠃⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠛⠿⢿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠻⢦⣀⠉⠉⠙⠛⠛⣉⣽⠟⠁⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⡀⠀⠀⠀⠉⠛⠲⠖⠚⠛⠉⠀⠀⠀⠀⣠⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠦⣤⣀⣀⣀⣀⣀⣀⣀⣠⡤⠶⠛⠉⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠶⠒⠒⠒⠒⠶⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠐⠦⠀⡀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡄⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⣀⠀⠀⠀⢸⣿⣅⠀⠀⠀⠀⣀⠀⢀⣬⣾⠇⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠉⠂⢋⠀⠀⡶⠾⠛⢛⣴⡋⠉⠉⢭⡍⠘⠛⠡⣆⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠘⠄⠀⠈⠀⠀⠀⠙⠀⠀⠀⠈⠁⠀⠀⢤⡼⣷⠆
⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠⢂⡀⠀⠠⠀⠀⠀⢀⣃⡉⡾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣨⠟⠃⡁⡄⡇⣇⠀⢏⠀⠀⠀⢨⣭⠀⠀⠀⢀⠐⣿⡿⠉⣤⣧
⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⢰⠃⠃⠁⠙⠃⡌⢰⠀⠀⠈⠛⠋⠀⡠⢚⢆⡍⠛⠛⢬⣅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⡈⢡⢸⢤⠀⠀⠠⠐⠋⠀⣸⠈⡏⢸⢦⢠⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠘⢼⡀⠀⠀⠀⢀⡾⠁⠀⡀⢀⡄⠇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠄⢠⡀⣠⠏⡠⠈⡇⠀⠀⠀⠜⢠⠀⠸⠁⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠘⠉⠘⠡⠂⠀⠀⠀⠀⠠⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠮⠂⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⢀⢠⠀⠀⢤
⠀⠀⠀⠀⡞⠧⠗⠒⠒⠚⠤⢳⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢀⠀⠈⢢⢤⠣⢺⠬⠊
⠀⠀⠀⢰⠉⠉⠉⣭⠉⠉⠉⣽⠀⠀⠀⠀⠀⠀⠀⠈⠣⣀⣀⠜⠁⠀⠀⠀⠀⠀⠀⠈⠈⠍⡪⠁⠀⠀⠨⣗⠉
⠀⠀⢀⣎⣀⣀⣐⣃⣀⣀⡰⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣹⠥⣠⣠⠼⡉⠉⠐
⠀⠀⠀⠀⠓⠒⠒⠒⠒⠒⠒⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠌⠁⠰⠁⠸⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢄⠀⡤⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠁
⠀⠀⠀⢀⠔⠊⠩⠍⠑⠢⡈⠐⠀⠀⠀⠀⠀⠀⢀⡤⢍⡊⠀⡨⠥⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡛⣖⡁⠁
⠀⠀⢀⠏⠐⠤⣐⡂⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⡿⠽⣒⡗⢸⣒⠯⠽⡀⠀⠀⠀⠀⠀⠀⠐⢂⣀⠤⠚⡶⢃⠀⡀
⠀⠀⠈⡍⠁⠀⠈⠁⠀⠈⢹⠀⠀⠀⠀⠀⠀⠸⡀⢀⡜⠀⠀⠣⡀⢀⡇⠀⠀⠀⠀⠀⢀⡔⢫⡀⠀⡞⠀⠈⠁
⠀⠀⢀⠘⢦⣀⠀⠄⣀⠴⠃⠀⠀⠀⠀⠀⣠⢄⡏⠁⠀⠀⠀⠀⠈⢹⡤⣄⠀⠀⠀⢰⢧⠙⢦⣩⠞
⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⠃⠀⠀⠀⠀⠘⠓⠚⠁⠀⠀⠀⠈⠳⠽⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⣀⠐⢀⣟⠁⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⠐⡦⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⢀⠀⢀
⠀⠀⠀⠺⠮⠄⡮⡼⡆⠒⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⣧⡀⠀⡇⠀⠀⠀⠀⠐⠆⢈⢄⡠⣄⡤⣀⢤⠰⢀⡀
⠀⠀⠀⠀⠂⡼⢇⡘⣹⡄⢻⠋⠀⠀⠀⠀⠀⠀⠒⢀⠜⡁⢇⣸⠞⢉⠀⠀⠀⠀⠀⠬⠀⢀⣞⢇⡜⣰⣹⣸⠀⠨⠁
⠀⠀⠀⠀⢰⡓⠎⣤⡋⣳⢄⡀⠀⠀⠀⠀⠀⠀⠀⢨⠦⣧⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⠀⢀⠀⢀⠀⠠⠀⠑
⠀⠀⠀⠀⠀⠙⠻⠭⢉⣀⠤⠏⠀⠀⠀⠀⠀⠈⠌⢩⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠤⠄⡤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠂⠀⠈⠉⠉⠒⠂⠄⡀⠀⠀⠀⡀⠤⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠑⠶⠚⠁⢀⣄⠐⠁⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠐⠲⣄⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⢰⠁⡌⠀⠀⠆⠀⠈⠉⠉⠉⠀⠉⢉⠇⠀⠐⡏⠀⠀⠀⠀⠙⠢⠤⠤⠤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣤⣤⡤⠼⢧⡇⠀⠀⣇⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⢀
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠘⠦⣀⣉⣁⡤⠚⠁⠀⠀⠀⠀⢈⡴
⠂⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⡄⠀⠀⠶⠉
⠐⠀⡱⠐⠒⠈⠀⠀⠀⠁⠒⠠⠤⣩⣐⠠⢄⣀⡙⠦⣄⣀⠀⠀⠀⠀⠀⣀⣠⠞⠁⠀⠀⢀⠠⢐⡠⠆⠑⢄⡀⠀⡀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⠥⣁⣉⡹⢛⣛⡻⠛⢛⡋⠀⠤⠤⠄⠒⠂⠉⠁⠀⠀⠀⠀⠉⠈⠃⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⢠⡄⢀⡣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠲⠶⠶⠤⠤⠴⠖⠋⠡⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠐⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠠⠀⠄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣠⣤⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠒⠂⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢀⠂⡐⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡌⠄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠁⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠘⠔⠁⠀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠦⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⢀⠄⠂⡁⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠐⠂⠄⡁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠄⠊⢁⡠⠴⠒⠋⠉⠉⠉⠁⠉⠉⠉⠉⠓⠒⠦⠤⣀⠒⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡡⠖⠚⠛⠓⠂⠀⠀⠀⠀⠐⠉⢉⣉⡛⠒⠄⠀⠀⠀⠀⠀⠀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠉⣙⡙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⠀⠀⠀⠀⠀⠀⠨⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⢠⠄⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡠⠀⠀⠀⠀⠋⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢐⡈⠤⠢⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠄⠂⠀⠀⠀⠀⠈⡐⠐⠄
⠀⢀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠤⢤⣀⣀⠀⠀⠀⠀⠑
⠀⠊⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠉⠓⢄⡀⠠
⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠡⡁
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠐⠀⠀⠀⠀⠀⠰⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠸⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠄⠀⠀⠀⠀⠀⢄⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠈⠢⠠⠈⠀⠀⡠⠀⠀⠀⠀⢠⠀⠀⢃⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⡈⠀⠀⠈⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠐⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃
⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠦⠤⠤⣤⡀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⠀⠈⢳⡀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⣾⠉⠁⠂⠀⠀⠄⠠⠼⣷⡄
⠀⠀⠀⠀⠀⠀⠀⣹⣄⡀⠀⠀⠀⠀⡀⣠⣿⠇⠼⢦
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣛⠋⠉⣛⣿⣿⣿⡿⣄⣀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣾⢿⣿⡿⠟⠃⠀⠈⠉⣩⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⣥⣴⣾⣷⣤⣤⡄⠐⢻⠁⠀⠀⠀⠀⠈⡁⠴⠛⣶⡆⣠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⢀⡁⣠⣾⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠐⡔⠀⠀⠀⠙⠛⠈⠈⠁⠒⠒⠐⠒⢂⠉⠁⢀⡀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⢸⠟⢻⣿⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠅⠀⠀⠀⠂⠀⣁⣠⣶⣒⠒⠉⠉⠛⠉⠉
⠀⠀⠀⠀⠀⠈⡀⠀⠸⣿⣿⠃⠀⠀⠀⠀⠐⣄⠀⡀⠀⠀⠀⠀⡴⠆⠀⢂⡤⠀⠠⠔⠉⢉⠝⠁
⠀⠀⠀⠀⠀⠀⠹⡀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠉⢋⡑⠀⠀⠠⠽⠢⢡⡦⠛⠂⠀⠀⣀⡊⠁
⠀⠀⠀⠀⠀⠀⠀⠹⡘⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠤⠘⠀⠄⠁⠀⠀⠀⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣧⣤⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⠮⠧⣦⣤⣴⣶⣾⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⢀⣀⡴⢞⣿⠧⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠛⠛⠙⠛⠛⠛⠛⠛⠙⠛⠙⠛⠛⠋⠓

⣿⢿⠿⡿⡿⢿⠿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣛⣟⣟⣿⣿⡿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠈⠉⠉⠀⠒⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠠⠄⠀⠄⣀⠀⠀⠀⢀⠔⠀⠉⣠⣀⣀⠤⠤⠤⣀⠀⠀⢸⣄⡀
⠀⠀⠀⠀⢀⠤⠒⠉⠉⠉⠉⠀⠀⠉⠑⠢⣀⠀⠀⠀⠀⠀⣠⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⠄⠁⣠⠖⠊⠁⠀⠀⠀⠀⠀⠈⠑⢄⠀⠀⠙⡆
⠀⠀⠀⠔⠁⢀⠶⠦⡀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢠⢂⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⡌
⠀⠀⠄⠠⠚⠁⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠈⡇⢀⠋⢠⡠⠤⠴⠒⠒⠒⠦⠤⣸⠀⠀⠀⠘⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢣
⠀⠀⡚⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠤⣀⠀⠀⠀⡇⡌⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⢸⡀⠀⠀⠀⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⡗⣦⠂
⠀⠀⠈⠀⠀⠀⠀⠐⠀⠀⠀⠠⠀⠀⠀⠙⠢⣤⠃⡇⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⣰⢓⡕
⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠛⠀⡅⠀⠸⡄⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠦⡷⣄⠀⠀⠠⠄⠀⡊⠀⠀⢀⠔⠁
⠰⠦⣀⠀⠑⠤⡀⠈⠂⠤⠤⠔⠁⢀⡠⠞⠁⣠⣴⠐⡄⢀⣘⣄⠈⠀⠐⠁⠺⢅⣀⠀⠀⠀⠀⢀⡜⠀⠈⠑⠲⠤⣤⠭⣤⡔⠚⠁
⠀⠀⠈⠙⡊⠒⡠⠋⢒⠒⠒⡖⠺⡥⠔⢲⠋⠀⢀⣀⣀⡀⡀⠀⢹⠒⠲⠶⡶⣶⠋⣉⠉⢉⡉⣩⡆⢀⣀⠂⢒⠎⢣⡀⡀⠈⢇⠒⣆⣀⣀⡀⡀
⠀⠀⠀⠀⠉⡝⠁⠀⠀⠀⠀⠀⠀⠹⡂⠊⠀⠀⠚⠃⠈⠉⠣⣀⠃⠀⠀⠈⠀⢸⣀⡠⠋⠉⢱⡄⠊⠉⠈⢆⠂⠀⠀⠀⠀⠀⠀⠳⡃⠀⠀⠉⠋
⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⡠⠴⠦⢈⡁⠒⠒⣦⠤⠤⠤⠀⠒⢞⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠸⣀⠀⠀⠠⠤⠤⠤⠤⠤⠤⢾⣀⡀
⠀⠀⠙⣆⠀⠀⠀⠉⠚⠁⠓⢠⠄⠐⠒⠚⠀⠀⠀⠀⠪⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡆⠀⠀⠀⠀⣆⣀⣀⣀⡀⢢⡠⠤⠔⠚⠁⡼⠁
⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⢶⠤⠤⠖⠒⠊⠱⣄⠀⣠⠄⠀⠀⢨⠃⠀⠀⠀⠈⠀⠀⠀⠀⠸⠷⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠈⠳⠃⠀⠓⢦⠎